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A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).

%I A281244 #25 Nov 24 2020 02:23:07
%S A281244 1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,1,1,0,0,0,0,2,1,0,0,0,1,
%T A281244 2,1,0,0,0,1,3,1,0,0,0,2,3,1,0,0,0,3,4,1,0,0,1,4,4,1,0,0,1,5,5,1,0,0,
%U A281244 2,7,5,1,0,0,3,8,6,1,0,0,5,10,6,1,0,1
%N A281244 Expansion of Product_{k>=1} (1 + x^(6*k-1)).
%C A281244 Convolution of this sequence and A280456 is A098884.
%H A281244 Vaclav Kotesovec, <a href="/A281244/b281244.txt">Table of n, a(n) for n = 0..10000</a>
%H A281244 Vaclav Kotesovec, <a href="/A281244/a281244_1.jpg">Graph - minor asymptotic term</a>, (Pi/144 - 9/(4*Pi))/sqrt(2) = -0.49100125...
%F A281244 a(n) ~ exp(sqrt(n/2)*Pi/3) / (2^(25/12)*sqrt(3)*n^(3/4)) * (1 + (Pi/144 - 9/(4*Pi)) / sqrt(2*n)).
%F A281244 G.f.: Sum_{k>=0} x^(k*(3*k + 2)) / Product_{j=1..k} (1 - x^(6*j)). - _Ilya Gutkovskiy_, Nov 24 2020
%p A281244 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A281244       [0$5, 1, 0$4, -1, 1][1+irem(d, 12)]*d, d=
%p A281244        numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A281244     end:
%p A281244 seq(a(n), n=0..100);  # _Alois P. Heinz_, Jan 18 2017
%t A281244 nmax = 200; CoefficientList[Series[Product[(1 + x^(6*k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
%t A281244 nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 6] == 5, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly
%Y A281244 Cf. A262928, A147599, A281243, A281245.
%Y A281244 Cf. A261612, A169975, A280454, A280456, A280457.
%K A281244 nonn
%O A281244 0,29
%A A281244 _Vaclav Kotesovec_, Jan 18 2017