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%I #23 Nov 14 2017 17:24:23
%S 1,1,5,17,44,127,332,866,2182,5412,13119,31292,73516,170136,388829,
%T 877653,1959111,4327221,9464856,20511598,44067446,93901142,198539477,
%U 416696608,868448305,1797890682,3698350956,7561361750,15369154555,31064311255,62449795986,124895635385,248538538858,492207649241
%N Expansion of Product_{k>=1} (1 + x^k)^(k*(3*k-1)/2).
%C Weigh transform of the pentagonal numbers (A000326).
%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(3*n-1)/2, g(n) = -1. - _Seiichi Manyama_, Nov 14 2017
%H Seiichi Manyama, <a href="/A294102/b294102.txt">Table of n, a(n) for n = 0..10000</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/PentagonalNumber.html">Pentagonal Number</a>
%F G.f.: Product_{k>=1} (1 + x^k)^A000326(k).
%F a(n) ~ exp(-225*Zeta(3)^3 / (98*Pi^8) - 9 * 5^(5/4) * Zeta(3)^2 / (4 * 7^(5/4) * Pi^5) * n^(1/4) - (3*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2 * (7/5)^(1/4) * Pi / 3) * n^(3/4)) * 7^(1/8) / (2^(47/24) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 10 2017
%F a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(3*d-1)*(-1)^(1+n/d). - _Seiichi Manyama_, Nov 14 2017
%t nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(k (3 k - 1)/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d^2 (3 d - 1)/2, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]
%Y Cf. A000326, A027998, A028377, A294836, A294837, A294838.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 09 2017
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