A295179 - OEIS

%I #8 Nov 16 2017 17:19:50

%S 1,1,5,15,44,115,312,790,2004,4908,11885,28170,65987,152079,346560,

%T 779808,1736460,3825995,8351733,18064545,38747740,82443251,174096564,

%U 364991008,759989218,1572126699,3231929735,6604498620,13419469596,27117216441,54508611399,109013531864,216956853105

%N Expansion of Product_{k>=1} 1/(1 - x^k)^(3*k*(k-1)/2+1).

%C Euler transform of the centered triangular numbers (A005448).

%H Vaclav Kotesovec, <a href="/A295179/b295179.txt">Table of n, a(n) for n = 0..4000</a>

%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CenteredTriangularNumber.html">Centered Triangular Number</a>

%H <a href="/index/Ce#CENTRALCUBE">Index entries for sequences related to centered polygonal numbers</a>

%F G.f.: Product_{k>=1} 1/(1 - x^k)^A005448(k).

%F a(n) ~ exp(-3*Zeta'(-1)/2 + 7*Zeta(3) / (8*Pi^2) - 225*Zeta(3)^3 / (2*Pi^8) + (Pi / (3*2^(3/4)) - 45*Zeta(3)^2 / (2^(7/4) * Pi^5)) * (5*n)^(1/4) - (3*sqrt(5/2) * Zeta(3) / Pi^2) * sqrt(n) + (2^(7/4)*Pi / (3*5^(1/4))) * n^(3/4)) / (2^(71/32) * 5^(7/32) * Pi^(1/8) * n^(23/32)). - _Vaclav Kotesovec_, Nov 16 2017

%t nmax = 32; CoefficientList[Series[Product[1/(1 - x^k)^(3 k (k - 1)/2 + 1), {k, 1, nmax}], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

%Y Cf. A000294, A000335, A005448, A295180.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 16 2017