Search: a294288 -id:a294288
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A294300
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Sum of the fifth powers of the parts in the partitions of n into two distinct parts.
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+10
6
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0, 0, 33, 244, 1300, 4182, 12201, 27984, 61776, 117700, 220825, 374100, 630708, 985194, 1539825, 2266432, 3347776, 4708584, 6657201, 9033300, 12333300, 16256350, 21571033, 27758544, 35970000, 45364332, 57617001, 71428084, 89176276, 108928050, 133987425
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).
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FORMULA
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a(n) = Sum_{i=1..floor(n/2)-((n+1) mod 2)} i^5 + (n-i)^5.
G.f.: -x^3*(33 +211*x +858*x^2 +1616*x^3 +2178*x^4 +1656*x^5 +858*x^6 +236*x^7 +33*x^8 +x^9) /(1+x)^6 /(x-1)^7. - R. J. Mathar, Nov 07 2017
a(n) = (1/192)*(n^2*(-16 + 80*n^2 - 3*(33 + (-1)^n)*n^3 + 32*n^4)).
a(n) = a(n-1) + 6*a(n-2) - 6*a(n-3) - 15*a(n-4) + 15*a(n-5) + 20*a(n-6) - 20*a(n-7) - 15*a(n-8) + 15*a(n-9) + 6*a(n-10) - 6*a(n-11) - a(n-12) + a(n-13) for n>13.
(End)
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MATHEMATICA
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Table[Sum[i^5 + (n - i)^5, {i, Floor[n/2] - Mod[n + 1, 2]}], {n, 40}]
Table[Total[Flatten[Select[IntegerPartitions[n, {2}], #[[1]]!=#[[2]]&]]^5], {n, 40}] (* Harvey P. Dale, Sep 04 2024 *)
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PROG
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(PARI) concat(vector(2), Vec(x^3*(33 + 211*x + 858*x^2 + 1616*x^3 + 2178*x^4 + 1656*x^5 + 858*x^6 + 236*x^7 + 33*x^8 + x^9) / ((1 - x)^7*(1 + x)^6) + O(x^40))) \\ Colin Barker, Nov 21 2017
(PARI) a(n) = sum(i=1, (n-1)\2, i^5 + (n-i)^5); \\ Michel Marcus, Nov 22 2017
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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STATUS
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approved
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A294301
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Sum of the sixth powers of the parts in the partitions of n into two distinct parts.
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+10
5
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0, 0, 65, 730, 4890, 19786, 67171, 180724, 446964, 962780, 1978405, 3703310, 6735950, 11445110, 19092295, 30220776, 47260136, 70866264, 105409929, 151455810, 216455810, 300450370, 415601835, 560651740, 754740700, 994054516, 1307797101, 1687688054, 2177107894
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^6 + (n-i)^6.
G.f.: x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7).
a(n) = (n/42 - n^3/6 + n^5/2 - 1/128*(65 + (-1)^n)*n^6 + n^7/7).
a(n) = a(n-1) + 7*a(n-2) - 7*a(n-3) - 21*a(n-4) + 21*a(n-5) + 35*a(n-6) - 35*a(n-7) - 35*a(n-8) + 35*a(n-9) + 21*a(n-10) - 21*a(n-11) - 7*a(n-12) + 7*a(n-13) + a(n-14) - a(n-15) for n>15.
(End)
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MATHEMATICA
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Table[Sum[i^6 + (n - i)^6, {i, Floor[(n-1)/2]}], {n, 40}]
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i^6 + (n-i)^6); \\ Michel Marcus, Nov 08 2017
(PARI) concat(vector(2), Vec(x^3*(65 + 665*x + 3705*x^2 + 10241*x^3 + 19630*x^4 + 23246*x^5 + 19630*x^6 + 10486*x^7 + 3705*x^8 + 721*x^9 + 65*x^10 + x^11) / ((1 - x)^8*(1 + x)^7) + O(x^40))) \\ Colin Barker, Nov 20 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A294302
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Sum of the seventh powers of the parts in the partitions of n into two distinct parts.
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+10
4
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0, 0, 129, 2188, 18700, 94638, 376761, 1183920, 3297456, 8002300, 18080425, 37287660, 73399404, 135324378, 241561425, 410323648, 680856256, 1086411960, 1703414961, 2587286700, 3877286700, 5658888070, 8172733129, 11541726768, 16164030000, 22204797108
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,8,-8,-28,28,56,-56,-70,70,56,-56,-28,28,8,-8,-1,1).
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^7 + (n-i)^7.
G.f.: x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8).
a(n) = (1/768)*(n^2*(64 - 224*n^2 + 448*n^4 - 3*(129 + (-1)^n)*n^5 + 96*n^6)).
a(n) = a(n-1) + 8*a(n-2) - 8*a(n-3) - 28*a(n-4) + 28*a(n-5) + 56*a(n-6) - 56*a(n-7) - 70*a(n-8) + 70*a(n-9) + 56*a(n-10) - 56*a(n-11) - 28*a(n-12) + 28*a(n-13) + 8*a(n-14) - 8*a(n-15) - a(n-16) + a(n-17) for n>17.
(End)
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MATHEMATICA
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Table[Sum[i^7 + (n - i)^7, {i, Floor[(n-1)/2]}], {n, 40}]
CoefficientList[Series[x^3(129+2059x+15480x^2+59466x^3+153639x^4+257307x^5+ 311664x^6+ 258532x^7+153639x^8+60537x^9+15480x^10+2178x^11+129x^12+x^13)/ ((1-x)^9 (1+x)^8), {x, 0, 60}], x] (* or *) LinearRecurrence[{1, 8, -8, -28, 28, 56, -56, -70, 70, 56, -56, -28, 28, 8, -8, -1, 1}, {0, 0, 0, 129, 2188, 18700, 94638, 376761, 1183920, 3297456, 8002300, 18080425, 37287660, 73399404, 135324378, 241561425, 410323648}, 60] (* Harvey P. Dale, Aug 05 2021 *)
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i^7 + (n-i)^7); \\ Michel Marcus, Nov 08 2017
(PARI) concat(vector(2), Vec(x^3*(129 + 2059*x + 15480*x^2 + 59466*x^3 + 153639*x^4 + 257307*x^5 + 311664*x^6 + 258532*x^7 + 153639*x^8 + 60537*x^9 + 15480*x^10 + 2178*x^11 + 129*x^12 + x^13) / ((1 - x)^9*(1 + x)^8) + O(x^40))) \\ Colin Barker, Nov 20 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A294303
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Sum of the eighth powers of the parts in the partitions of n into two distinct parts.
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+10
3
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0, 0, 257, 6562, 72354, 456418, 2142595, 7841860, 24684612, 67340708, 167731333, 380410598, 812071910, 1622037830, 3103591687, 5649705096, 9961449608, 16894160328, 27957167625, 44840730666, 70540730666, 108149231146, 163239463563, 241120467148, 351625763020
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-36,36,84,-84,-126,126,126,-126,-84,84,36,-36,-9,9,1,-1).
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^8 + (n-i)^8.
For odd n, a(n) = n^9 / 9 - n^8/2 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30
For even n, a(n) = n^9 / 9 - 129*n^8/256 + 2*n^7 / 3 - 7*n^5 / 15 + 2*n^3 / 9 - n/30.
(End)
G.f.: x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9).
a(n) = a(n-1) + 9*a(n-2) - 9*a(n-3) - 36*a(n-4) + 36*a(n-5) + 84*a(n-6) - 84*a(n-7) - 126*a(n-8) + 126*a(n-9) + 126*a(n-10) - 126*a(n-11) - 84*a(n-12) + 84*a(n-13) + 36*a(n-14) - 36*a(n-15) - 9*a(n-16) + 9*a(n-17) + a(n-18) - a(n-19) for n>19.
(End)
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MATHEMATICA
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Table[Sum[i^8 + (n - i)^8, {i, Floor[(n-1)/2]}], {n, 30}]
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PROG
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(PARI) first(n) = {my(res = vector(n, i, 1/9*i^9 - 1/2*i^8 + 2/3*i^7 - 7/15*i^5 + 2/9*i^3 - 1/30*i)); forstep(i = 2, #res, 2, res[i] -= i^8/256); res} \\ David A. Corneth, Nov 05 2017
(PARI) concat(vector(2), Vec(x^3*(257 + 6305*x + 63479*x^2 + 327319*x^3 + 1103301*x^4 + 2469669*x^5 + 4014083*x^6 + 4659395*x^7 + 4014083*x^8 + 2480995*x^9 + 1103301*x^10 + 331365*x^11 + 63479*x^12 + 6551*x^13 + 257*x^14 + x^15) / ((1 - x)^10*(1 + x)^9) + O(x^40))) \\ Colin Barker, Nov 20 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A294304
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Sum of the ninth powers of the parts of the partitions of n into two distinct parts.
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+10
2
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0, 0, 513, 19684, 282340, 2215782, 12313161, 52404624, 186884496, 572351860, 1574304985, 3922174980, 9092033028, 19656178794, 40357579185, 78666720832, 147520415296, 265720871304, 464467582161, 786155279940, 1299155279940, 2091077378830, 3300704544313
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,10,-10,-45,45,120,-120,-210,210,252,-252,-210,210,120,-120,-45,45,10,-10,-1,1).
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.
G.f.: x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10).
a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) - 45*a(n-4) + 45*a(n-5) + 120*a(n-6) - 120*a(n-7) - 210*a(n-8) + 210*a(n-9) + 252*a(n-10) - 252*a(n-11) - 210*a(n-12) + 210*a(n-13) + 120*a(n-14) - 120*a(n-15) - 45*a(n-16) + 45*a(n-17) + 10*a(n-18) - 10*a(n-19) - a(n-20) + a(n-21) for n>21.
(End)
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MATHEMATICA
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Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}]
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i^9 + (n-i)^9); \\ Michel Marcus, Nov 08 2017
(PARI) concat(vector(2), Vec(x^3*(513 + 19171*x + 257526*x^2 + 1741732*x^3 + 7493904*x^4 + 21619738*x^5 + 45264042*x^6 + 69257104*x^7 + 80125470*x^8 + 69325060*x^9 + 45264042*x^10 + 21693364*x^11 + 7493904*x^12 + 1755838*x^13 + 257526*x^14 + 19672*x^15 + 513*x^16 + x^17) / ((1 - x)^11*(1 + x)^10) + O(x^40))) \\ Colin Barker, Nov 20 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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A294305
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Sum of the tenth powers of the parts in the partitions of n into two distinct parts.
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+10
1
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0, 0, 1025, 59050, 1108650, 10815226, 71340451, 352767124, 1427557524, 4904576300, 14914341925, 40791300350, 102769130750, 240345147350, 529882277575, 1105458926376, 2206044295976, 4218551412024, 7792505423049, 13913571680850, 24163571680850, 40817515234450
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OFFSET
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1,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-55,55,165,-165,-330,330,462,-462,-462,462,330,-330,-165,165,55,-55,-11,11,1,-1).
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FORMULA
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a(n) = Sum_{i=1..floor((n-1)/2)} i^10 + (n-i)^10.
G.f.: x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11).
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 55*a(n-4) + 55*a(n-5) + 165*a(n-6) - 165*a(n-7) - 330*a(n-8) + 330*a(n-9) + 462*a(n-10) - 462*a(n-11) - 462*a(n-12) + 462*a(n-13) + 330*a(n-14) - 330*a(n-15) - 165*a(n-16) + 165*a(n-17) + 55*a(n-18) - 55*a(n-19) - 11*a(n-20) + 11*a(n-21) + a(n-22) - a(n-23) for n>23.
(End)
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MATHEMATICA
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Table[Sum[i^10 + (n - i)^10, {i, Floor[(n-1)/2]}], {n, 30}]
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PROG
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(PARI) a(n) = sum(i=1, (n-1)\2, i^10 + (n-i)^10); \\ Michel Marcus, Nov 05 2017
(PARI) concat(vector(2), Vec(x^3*(1025 + 58025*x + 1038325*x^2 + 9068301*x^3 + 49036000*x^4 + 177845712*x^5 + 466571800*x^6 + 905612928*x^7 + 1343112850*x^8 + 1525782114*x^9 + 1343112850*x^10 + 906468090*x^11 + 466571800*x^12 + 178253064*x^13 + 49036000*x^14 + 9115128*x^15 + 1038325*x^16 + 59037*x^17 + 1025*x^18 + x^19) / ((1 - x)^12*(1 + x)^11) + O(x^40))) \\ Colin Barker, Nov 21 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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