Search: a292193 -id:a292193
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A265837
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Expansion of Product_{k>=1} 1/(1 - k^3*x^k).
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+10
7
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1, 1, 9, 36, 164, 505, 2474, 7273, 31008, 103644, 379890, 1226802, 4747529, 14553648, 52167558, 171639695, 583371802, 1851395692, 6427705062, 19983302144, 67235043192, 214615427776, 697704303005, 2194982897304, 7262755260410, 22402942281766, 72461661415093
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * 3^n, where
c = 86.60286320343345379122228784466307940393110978... if n mod 3 = 0
c = 86.27536745612304663727011387030370600864018892... if n mod 3 = 1
c = 86.29819842537784019895326532818285333403267092... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(3*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[1/(1 - k^3*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A265838
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Expansion of Product_{k>=1} 1/(1 - k^4*x^k).
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+10
5
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1, 1, 17, 98, 610, 2531, 18580, 72453, 449494, 2114440, 10753594, 48572844, 272867295, 1137441506, 5834448870, 27276382027, 129389072144, 576677550870, 2884567552542, 12401875640710, 59474089385344, 270438887909580, 1230979340265033, 5477371267093144
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * 3^(4*n/3), where
c = 27.2472595510480930563087281042486261391960582835336715327... if n mod 3 = 0
c = 26.8841208067599453033952496040472485838861626762931432887... if n mod 3 = 1
c = 26.9277867007233095885556073185206409643421012262073908850... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(4*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[1/(1 - k^4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A265839
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Expansion of Product_{k>=1} 1/(1 - k^5*x^k).
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+10
5
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1, 1, 33, 276, 2324, 13225, 145586, 760057, 6836328, 45996924, 322816122, 2064921330, 16881567137, 96217644312, 708147553326, 4769313137735, 31412238427954, 198869428043476, 1442034056253438, 8596120396405880, 58954590481229064, 387170921610808720
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) ~ c * 3^(5*n/3), where
c = 12.8519823810391431573687005461910113782018563173082562291... if n mod 3 = 0
c = 12.4535903496941652158697054030067622653283880393322526099... if n mod 3 = 1
c = 12.5138855694494734654940524026530463555984202132997900068... if n mod 3 = 2.
G.f.: exp(Sum_{k>=1} Sum_{j>=1} j^(5*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 14 2018
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MATHEMATICA
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nmax = 40; CoefficientList[Series[Product[1/(1 - k^5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A292194
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Sum of n-th powers of products of terms in all partitions of n.
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+10
5
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1, 1, 5, 36, 610, 13225, 1173652, 92137513, 27960729094, 14612913824364, 11885159817456154, 23676862215173960082, 144210774157588042096815, 778807208565930895328294712, 15863318347221014170216633451982, 908978343753718115412387406378667615
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = [x^n] Product_{k=1..n} 1/(1 - k^n*x^k).
a(n) ~ 3^(n^2/3) if mod(n,3)=0
a(n) ~ 3^(n*(n-4)/3)*2^(2*n+1) if mod(n,3)=1
a(n) ~ 3^(n*(n-2)/3)*2^n if mod(n,3)=2
(End)
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EXAMPLE
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5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1.
So a(5) = 5^5 + (4*1)^5 + (3*2)^5 + (3*1*1)^5 + (2*2*1)^5 + (2*1*1*1)^5 + (1*1*1*1*1)^5 = 13225.
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MAPLE
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b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
`if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
end:
a:= n-> b(n$3):
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MATHEMATICA
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nmax = 20; Table[SeriesCoefficient[Product[1/(1 - k^n*x^k), {k, 1, n}], {x, 0, n}], {n, 0, nmax}] (* Vaclav Kotesovec, Sep 15 2017 *)
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PROG
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(PARI) {a(n) = polcoeff(1/prod(k=1, n, 1-k^n*x^k+x*O(x^n)), n)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A294579
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Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(1 + k*n/d).
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+10
5
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1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 25, 6, 1, 33, 82, 97, 26, 12, 1, 65, 244, 385, 126, 80, 8, 1, 129, 730, 1537, 626, 588, 50, 15, 1, 257, 2188, 6145, 3126, 4508, 344, 161, 13, 1, 513, 6562, 24577, 15626, 35652, 2402, 2049, 163, 18
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OFFSET
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1,3
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LINKS
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FORMULA
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L.g.f. of column k: -log(Product_{j>=1} (1 - j^k * x^j)). - Seiichi Manyama, Jun 02 2019
G.f. of column k: Sum_{j>0} j^(k+1) * x^j / (1 - j^k * x^j). - Seiichi Manyama, Jan 14 2023
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, ...
3, 5, 9, 17, 33, ...
4, 10, 28, 82, 244, ...
7, 25, 97, 385, 1537, ...
6, 26, 126, 626, 3126, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A294582
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Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j)^j.
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+10
2
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1, 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 9, 14, 13, 1, 1, 17, 36, 42, 24, 1, 1, 33, 98, 148, 103, 48, 1, 1, 65, 276, 546, 489, 289, 86, 1, 1, 129, 794, 2068, 2467, 1959, 690, 160, 1, 1, 257, 2316, 7962, 12969, 14281, 6326, 1771, 282, 1, 1, 513, 6818, 30988, 70243
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OFFSET
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0,6
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LINKS
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FORMULA
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A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, ...
3, 5, 9, 17, 33, ...
6, 14, 36, 98, 276, ...
13, 42, 148, 546, 2068, ...
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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A292068
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Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 + j^k*x^j).
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+10
1
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1, 1, -1, 1, -1, 0, 1, -1, -1, -1, 1, -1, -3, -2, 1, 1, -1, -7, -6, 2, -1, 1, -1, -15, -20, 6, -1, 1, 1, -1, -31, -66, 20, 5, 4, -1, 1, -1, -63, -212, 66, 71, 40, -1, 2, 1, -1, -127, -666, 212, 605, 442, 11, 18, -2, 1, -1, -255, -2060, 666, 4439, 4660, 215, 226, -22, 2
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OFFSET
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0,13
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LINKS
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, ...
-1, -1, -1, -1, -1, ...
0, -1, -3, -7, -15, ...
-1, -2, -6, -20, -66, ...
1, 2, 6, 20, 66, ...
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MAPLE
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b:= proc(n, i, k) option remember; (m->
`if`(m<n, 0, `if`(n=m, i!^k, b(n, i-1, k)+
`if`(i>n, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
end:
A:= proc(n, k) option remember; `if`(n=0, 1,
-add(b(n-i$2, k)*A(i, k), i=0..n-1))
end:
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MATHEMATICA
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b[n_, i_, k_] := b[n, i, k] = If[# < n, 0, If[n == #, i!^k, b[n, i-1, k] + If[i > n, 0, i^k b[n-i, i-1, k]]]]&[i(i+1)/2];
A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[b[n-i, n-i, k] A[i, k], {i, 0, n-1}]];
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PROG
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(Python)
from sympy.core.cache import cacheit
from sympy import factorial as f
@cacheit
def b(n, i, k):
m=i*(i + 1)/2
return 0 if m<n else f(i)**k if n==m else b(n, i - 1, k) + (0 if i>n else i**k*b(n - i, i - 1, k))
@cacheit
def A(n, k): return 1 if n==0 else -sum([b(n - i, n - i, k)*A(i, k) for i in range(n)])
for d in range(13): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Sep 14 2017, after Maple program
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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