A294958 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*((k-2)^2+k)/2).
1, 1, 3, 9, 28, 75, 198, 494, 1243, 3061, 7500, 18055, 43057, 101292, 236178, 545218, 1248480, 2835059, 6390360, 14298631, 31778782, 70168935, 153993321, 335977369, 728962258, 1573189113, 3377881482, 7217395643, 15348900996, 32494548816, 68494383520, 143773075158, 300568066729
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
- M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
- N. J. A. Sloane, Transforms
- Index to sequences related to polygonal numbers
Programs
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Maple
N:=100: S:= series(mul(1/(1 - x^k)^(k*((k-2)^2+k)/2),k=1..N),x,N+1): seq(coeff(S,x,k),k=0..N); # Robert Israel, Nov 12 2017
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Mathematica
nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(k ((k - 2)^2 + k)/2), {k, 1, nmax}], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 ((d - 2)^2 + d)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^A060354(k).
a(n) ~ exp(2*Zeta'(-1) + 3*Zeta(3) / (8*Pi^2) - Pi^16 / (1036800000 * Zeta(5)^3) + Pi^8 * Zeta(3) / (36000 * Zeta(5)^2) - 2*Zeta(3)^2 / (15*Zeta(5)) + Zeta'(-3)/2 + (-Pi^12 / (3600000 * 2^(2/5) * 3^(1/5) * Zeta(5)^(11/5)) + Pi^4 * Zeta(3) / (150 * 2^(2/5) * 3^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-Pi^8 / (12000 * 2^(4/5) * 3^(2/5) * Zeta(5)^(7/5)) + 2^(1/5) * Zeta(3) / (3*Zeta(5))^(2/5)) * n^(2/5) - (Pi^4 / (60 * 2^(1/5) * (3*Zeta(5))^(3/5))) * n^(3/5) + (5*(3*Zeta(5))^(1/5) / 2^(8/5)) * n^(4/5)) * (3*Zeta(5))^(53/400) / (2^(47/200) * sqrt(5*Pi) * n^(253/400)). - Vaclav Kotesovec, Nov 12 2017
Comments