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A261569 Expansion of Product_{k>=1} (1 + 5*x^k).

%I A261569 #21 Aug 23 2018 08:00:47
%S A261569 1,5,5,30,30,55,180,205,330,480,1230,1380,2255,3030,4530,8555,10680,
%T A261569 15330,21330,29730,39480,67380,81505,116280,153030,210930,270805,
%U A261569 370080,534330,675480,900480,1180380,1544130,1997280,2597280,3304805,4581180,5653080
%N A261569 Expansion of Product_{k>=1} (1 + 5*x^k).
%C A261569 In general, for a fixed integer m > 0, if g.f. = Product_{k>=1} (1 + m*x^k) then a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt((m+1)*Pi)*n^(3/4)), where c = Pi^2/6 + log(m)^2/2 + polylog(2, -1/m) = -polylog(2, -m). - _Vaclav Kotesovec_, Jan 04 2016
%H A261569 Vaclav Kotesovec, <a href="/A261569/b261569.txt">Table of n, a(n) for n = 0..10000</a>
%F A261569 a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(6*Pi)*n^(3/4)), where c = Pi^2/6 + log(5)^2/2 + polylog(2, -1/5) = 2.74927912606080829002558751537626864449... . - _Vaclav Kotesovec_, Jan 04 2016
%F A261569 G.f.: Sum_{i>=0} 5^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - _Ilya Gutkovskiy_, Apr 12 2018
%p A261569 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A261569       `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 5*b(n-i, i-1))))
%p A261569     end:
%p A261569 a:= n-> b(n$2):
%p A261569 seq(a(n), n=0..60);  # _Alois P. Heinz_, Aug 24 2015
%t A261569 nmax = 40; CoefficientList[Series[Product[1 + 5*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A261569 nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*5^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 25 2015 *)
%t A261569 (QPochhammer[-5, x]/6 + O[x]^58)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2015 *)
%Y A261569 Cf. A000009, A032302, A032308, A261568, A291698.
%K A261569 nonn
%O A261569 0,2
%A A261569 _Vaclav Kotesovec_, Aug 24 2015