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Search: a294288 -id:a294288
Displaying 1-6 of 6 results found. page 1
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A294300 Sum of the fifth powers of the parts in the partitions of n into two distinct parts. +10
6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,6,-6,-15,15,20,-20,-15,15,6,-6,-1,1).
FORMULA
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
From Colin Barker, Nov 21 2017: (Start)
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288.
KEYWORD
nonn,easy,changed
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
A294301 Sum of the sixth powers of the parts in the partitions of n into two distinct parts. +10
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
Index entries for linear recurrences with constant coefficients, signature (1,7,-7,-21,21,35,-35,-35,35,21,-21,-7,7,1,-1).
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} i^6 + (n-i)^6.
From Colin Barker, Nov 20 2017: (Start)
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)
MATHEMATICA
Table[Sum[i^6 + (n - i)^6, {i, Floor[(n-1)/2]}], {n, 40}]
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288, A294300.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
A294302 Sum of the seventh powers of the parts in the partitions of n into two distinct parts. +10
4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
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).
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} i^7 + (n-i)^7.
From Colin Barker, Nov 20 2017: (Start)
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)
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288, A294300, A294301.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
A294303 Sum of the eighth powers of the parts in the partitions of n into two distinct parts. +10
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
David A. Corneth, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions
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).
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} i^8 + (n-i)^8.
From David A. Corneth, Nov 05 2017: (Start)
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)
From Colin Barker, Nov 20 2017: (Start)
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)
MATHEMATICA
Table[Sum[i^8 + (n - i)^8, {i, Floor[(n-1)/2]}], {n, 30}]
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288, A294300, A294301, A294302.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
A294304 Sum of the ninth powers of the parts of the partitions of n into two distinct parts. +10
2
0, 0, 513, 19684, 282340, 2215782, 12313161, 52404624, 186884496, 572351860, 1574304985, 3922174980, 9092033028, 19656178794, 40357579185, 78666720832, 147520415296, 265720871304, 464467582161, 786155279940, 1299155279940, 2091077378830, 3300704544313 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
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).
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} i^9 + (n-i)^9.
From Colin Barker, Nov 20 2017: (Start)
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)
MATHEMATICA
Table[Sum[i^9 + (n - i)^9, {i, Floor[(n-1)/2]}], {n, 30}]
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
A294305 Sum of the tenth powers of the parts in the partitions of n into two distinct parts. +10
1
0, 0, 1025, 59050, 1108650, 10815226, 71340451, 352767124, 1427557524, 4904576300, 14914341925, 40791300350, 102769130750, 240345147350, 529882277575, 1105458926376, 2206044295976, 4218551412024, 7792505423049, 13913571680850, 24163571680850, 40817515234450 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for sequences related to partitions
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).
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} i^10 + (n-i)^10.
From Colin Barker, Nov 21 2017: (Start)
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)
MATHEMATICA
Table[Sum[i^10 + (n - i)^10, {i, Floor[(n-1)/2]}], {n, 30}]
PROG
(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
CROSSREFS
Cf. A294286, A294287, A294288, A294300, A294301, A294302, A294303, A294304.
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 27 2017
STATUS
approved
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Last modified September 11 14:25 EDT 2024. Contains 375836 sequences. (Running on oeis4.)