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%I #25 Aug 03 2019 05:27:59
%S 1,1,4,13,29,81,188,456,1030,2405,5295,11611,25246,53552,113332,
%T 235685,486011,990840,2006567,4018010,7992003,15768511,30875424,
%U 60060509,116042548,222817961,425200270,806991037,1522748592,2858792520,5339457208,9924370365
%N Expansion of Product_{k>=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="/A266891/b266891.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..5000 from Vaclav Kotesovec)
%H Vaclav Kotesovec, <a href="/A266891/a266891.jpg">Graph - The asymptotic ratio (200000 terms)</a>
%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*(-d)^(1+n/d). - _Seiichi Manyama_, Nov 18 2017
%F Conjecture: log(a(n)) ~ n^(2/3) * (2*log(3*n) - 3) / (4*3^(1/3)). - _Vaclav Kotesovec_, May 08 2018
%t nmax=50; CoefficientList[Series[Product[(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t (* 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 *)
%Y Cf. A022629, A026007, A032302, A261562, A266964, A304210, A304211.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, Jan 05 2016
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