A317017 Expansion of Product_{k>=1} 1/(1 - x^k)^((3*k+1)*binomial(k+2,3)/4).
1, 1, 8, 33, 126, 441, 1571, 5338, 17900, 58359, 187134, 588966, 1826537, 5580784, 16831549, 50135506, 147650112, 430187724, 1240908651, 3545808444, 10042128414, 28201458999, 78567720054, 217225969695, 596254164090, 1625343030654, 4401332943214, 11843216471115, 31674767502610
Offset: 0
Keywords
Links
- N. J. A. Sloane, Transforms
Programs
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Maple
a:=series(mul(1/(1-x^k)^((3*k+1)*binomial(k+2,3)/4),k=1..100),x=0,29): seq(coeff(a,x,n),n=0..28); # Paolo P. Lava, Apr 02 2019
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Mathematica
nmax = 28; CoefficientList[Series[Product[1/(1 - x^k)^((3 k + 1) Binomial[k + 2, 3]/4), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 28; CoefficientList[Series[Exp[Sum[x^k (1 + 2 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x] a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 (d + 1) (d + 2) (3 d + 1)/24, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 28}]
Formula
G.f.: Product_{k>=1} 1/(1 - x^k)^A001296(k).
G.f.: exp(Sum_{k>=1} x^k*(1 + 2*x^k)/(k*(1 - x^k)^5)).
a(n) ~ Pi^(1/288)/(2 * 3^(577/864) * 7^(145/1728) * n^(1009/1728)) * exp(1/144 - (1/12-Zeta'(-1))/12 - (11 * Zeta(3))/(80 * Pi^2) + (1383 * Zeta(5))/(640 * Pi^4) + (11025 * Zeta(3) * Zeta(5)^2)/(2 * Pi^12) - (694575 * Zeta(5)^3)/(2 * Pi^14) + (13127467500 * Zeta(5)^5)/Pi^24 + (5 * Zeta'(-3))/12 + ((-21 * 3^(1/3) * 7^(1/6) * Pi)/6400 - (35 * 3^(1/3) * 7^(1/6) * Zeta(3) * Zeta(5))/(2 * Pi^7) + (15435 * 3^(1/3) * 7^(1/6) * Zeta(5)^2)/(16 * Pi^9) - (175573125 * 3^(1/3) * 7^(1/6) * Zeta(5)^4)/(4 * Pi^19)) * n^(1/6) + (((7/3)^(1/3) * Zeta(3))/(4 * Pi^2) - (21 * 3^(2/3) * 7^(1/3) * Zeta(5))/(8 * Pi^4) + (147000 * 3^(2/3) * 7^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7) * Pi)/40 - (1575 * sqrt(7) * Zeta(5)^2)/Pi^9) * sqrt(n) + ((15 * 3^(1/3) * 7^(2/3) * Zeta(5))/(2 * Pi^4)) * n^(2/3) + ((2 * 3^(2/3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - Vaclav Kotesovec, Jul 28 2018
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