A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

1, 1, 4, 13, 29, 81, 188, 456, 1030, 2405, 5295, 11611, 25246, 53552, 113332, 235685, 486011, 990840, 2006567, 4018010, 7992003, 15768511, 30875424, 60060509, 116042548, 222817961, 425200270, 806991037, 1522748592, 2858792520, 5339457208, 9924370365

Offset: 0

Author

Vaclav Kotesovec, Jan 05 2016

Keywords

• nonn

Comments

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
• Vaclav Kotesovec, Graph - The asymptotic ratio (200000 terms)

Crossrefs

Cf. A022629, A026007, A032302, A261562, A266964, A304210, A304211.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 50; s = 1+x; Do[s*=Sum[Binomial[k, j] * k^j * x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 07 2016 *)

Formula

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(-d)^(1+n/d). - Seiichi Manyama, Nov 18 2017

Conjecture: log(a(n)) ~ n^(2/3) * (2*log(3*n) - 3) / (4*3^(1/3)). - Vaclav Kotesovec, May 08 2018