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A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.

%I A341365 #13 Feb 21 2021 02:12:03
%S A341365 1,8,40,156,520,1552,4262,10960,26716,62276,139744,303412,640001,
%T A341365 1315832,2644004,5204044,10052182,19086348,35672516,65708116,
%U A341365 119409576,214289116,380068582,666723748,1157550524,1990230968,3390558072,5726064688,9590759624,15938198484,26289242026
%N A341365 Expansion of (1 / theta_4(x) - 1)^4 / 16.
%H A341365 Robert Israel, <a href="/A341365/b341365.txt">Table of n, a(n) for n = 4..10000</a>
%F A341365 G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.
%F A341365 a(n) ~ A284286(n)/16. - _Vaclav Kotesovec_, Feb 20 2021
%p A341365 g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
%p A341365       g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
%p A341365     end:
%p A341365 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
%p A341365       g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A341365     end:
%p A341365 a:= n-> b(n, 4):
%p A341365 seq(a(n), n=4..34);  # _Alois P. Heinz_, Feb 10 2021
%t A341365 nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
%t A341365 nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &
%Y A341365 Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, A341366, A341367, A341368, A341369, A341370.
%K A341365 nonn
%O A341365 4,2
%A A341365 _Ilya Gutkovskiy_, Feb 10 2021