A295180 - OEIS

%I #14 Nov 17 2017 00:49:01

%S 1,1,4,14,35,96,242,609,1483,3565,8376,19389,44254,99584,221470,

%T 486810,1058914,2280519,4866492,10294313,21598679,44966391,92930485,

%U 190721585,388828094,787710401,1586166758,3175548134,6322372729,12520759979,24669499432,48367447687,94381633962,183331308393

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

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

%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -(3*n*(n-1)/2+1), g(n) = -1. - _Seiichi Manyama_, Nov 16 2017

%H Seiichi Manyama, <a href="/A295180/b295180.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/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 + x^k)^A005448(k).

%F a(n) ~ exp(15*Zeta(3) / (28*Pi^2) - 6075*Zeta(3)^3 / (98*Pi^8) + (Pi/6 - 405*Zeta(3)^2 / (28*Pi^5)) * (5*n/7)^(1/4) - (9*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2*Pi * (7/5)^(1/4)/3) * n^(3/4)) * 7^(1/8) / (2^(19/8) * 5^(1/8) * n^(5/8)). - _Vaclav Kotesovec_, Nov 16 2017

%F a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(3*d*(d-1)/2+1)*(-1)^(1+n/d). - _Seiichi Manyama_, Nov 16 2017

%t nmax = 33; CoefficientList[Series[Product[(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[(-1)^(k/d + 1) d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

%Y Cf. A005448, A028377, A258343, A295179.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 16 2017