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A341369 Expansion of (1 / theta_4(x) - 1)^8 / 256.

%I A341369 #9 Feb 10 2021 12:21:33
%S A341369 1,16,144,952,5136,23904,99292,376512,1324376,4372632,13673888,
%T A341369 40787848,116713350,321861312,858693192,2223428224,5602833292,
%U A341369 13772292360,33089930724,77846837848,179602530648,406914172336,906438716196,1987418937952,4293164981849,9144987747024
%N A341369 Expansion of (1 / theta_4(x) - 1)^8 / 256.
%H A341369 Alois P. Heinz, <a href="/A341369/b341369.txt">Table of n, a(n) for n = 8..10000</a>
%F A341369 G.f.: (1/256) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^8.
%p A341369 g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
%p A341369       g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
%p A341369     end:
%p A341369 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
%p A341369       g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A341369     end:
%p A341369 a:= n-> b(n, 8):
%p A341369 seq(a(n), n=8..33);  # _Alois P. Heinz_, Feb 10 2021
%t A341369 nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^8/256, {x, 0, nmax}], x] // Drop[#, 8] &
%t A341369 nmax = 33; CoefficientList[Series[(1/256) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^8, {x, 0, nmax}], x] // Drop[#, 8] &
%Y A341369 Cf. A002448, A004409, A014968, A015128, A319553, A327386, A338223, A340915, A341227, A341364, A341365, A341366, A341367, A341368, A341370.
%K A341369 nonn
%O A341369 8,2
%A A341369 _Ilya Gutkovskiy_, Feb 10 2021