A266941 - OEIS

%I #27 Aug 27 2018 05:46:31

%S 1,1,5,14,42,103,289,690,1771,4206,10142,23449,54786,123528,279480,

%T 619206,1366405,2969071,6425534,13727775,29187555,61439660,128620370,

%U 267044222,551527679,1130806020,2306746335,4676096006,9432394144,18920266428,37776372312

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

%C This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = n, g(n) = n. - _Seiichi Manyama_, Nov 18 2017

%H Seiichi Manyama, <a href="/A266941/b266941.txt">Table of n, a(n) for n = 0..6086</a> (terms 0..1000 from Vaclav Kotesovec)

%F From _Vaclav Kotesovec_, Jan 08 2016: (Start)

%F a(n) ~ c * n^2 * 3^(n/3), where

%F c = 3278974684037157122864203.021982619109776972432419491714093... if mod(n,3)=0

%F c = 3278974684037157122864202.999526122508793149896683112820555... if mod(n,3)=1

%F c = 3278974684037157122864203.001231135511323719311281438384212... if mod(n,3)=2

%F (End)

%F In closed form, a(n) ~ (Product_{k>=4}((1 - k/3^(k/3))^(-k)) / ((1 - 2/3^(2/3))^2 * (1 - 1/3^(1/3))) + Product_{k>=4}((1 - (-1)^(2*k/3)*k/3^(k/3))^(-k)) / ((-1)^(2*n/3) * ((1 + 2*(-1)^(1/3)/3^(2/3))^2 * (1 - (-1)^(2/3)/3^(1/3)))) + Product_{k>=4}((1 - (-1)^(4*k/3)*k/3^(k/3))^(-k)) / ((-1)^(4*n/3) * ((1 + (-1/3)^(1/3)) * (1 - 2*(-1/3)^(2/3))^2))) * 3^(n/3) * n^2 / 54. - _Vaclav Kotesovec_, Apr 24 2017

%F a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (Sum_{d|k} d^(2+k/d)) * a(n-k) for n > 0. - _Seiichi Manyama_, Nov 02 2017

%t nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 40; s = 1 - x; Do[s *= Sum[Binomial[k, j]*(-1)^j*k^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2018 *)

%Y Cf. A006906, A265758, A266891, A266942, A266964, A266971.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Jan 06 2016