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A341368 Expansion of (1 / theta_4(x) - 1)^7 / 128.

%I A341368 #9 Feb 10 2021 11:03:32
%S A341368 1,14,112,665,3248,13776,52437,183080,595399,1824109,5310144,14787542,
%T A341368 39605363,102465972,257005641,626841236,1490521109,3462881324,
%U A341368 7875519169,17562223791,38456245849,82793422502,175452110162,366348547908,754392685046,1533283745644,3078157040665
%N A341368 Expansion of (1 / theta_4(x) - 1)^7 / 128.
%H A341368 Alois P. Heinz, <a href="/A341368/b341368.txt">Table of n, a(n) for n = 7..10000</a>
%F A341368 G.f.: (1/128) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^7.
%p A341368 g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
%p A341368       g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
%p A341368     end:
%p A341368 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
%p A341368       g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A341368     end:
%p A341368 a:= n-> b(n, 7):
%p A341368 seq(a(n), n=7..33);  # _Alois P. Heinz_, Feb 10 2021
%t A341368 nmax = 33; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^7/128, {x, 0, nmax}], x] // Drop[#, 7] &
%t A341368 nmax = 33; CoefficientList[Series[(1/128) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^7, {x, 0, nmax}], x] // Drop[#, 7] &
%Y A341368 Cf. A002448, A004408, A014968, A015128, A327385, A338223, A340906, A341226, A341364, A341365, A341366, A341367, A341369, A341370.
%K A341368 nonn
%O A341368 7,2
%A A341368 _Ilya Gutkovskiy_, Feb 10 2021