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A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.

%I A338223 #11 Feb 10 2021 11:41:30
%S A338223 1,4,12,30,68,144,289,556,1034,1868,3292,5678,9608,15984,26188,42314,
%T A338223 67509,106460,166090,256552,392628,595696,896484,1338894,1985298,
%U A338223 2923840,4278448,6222518,8997544,12938368,18507297,26340040,37307326,52597320,73825504,103180702
%N A338223 G.f.: (1 / theta_4(x) - 1)^2 / 4, where theta_4() is the Jacobi theta function.
%H A338223 Alois P. Heinz, <a href="/A338223/b338223.txt">Table of n, a(n) for n = 2..10000</a>
%F A338223 G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.
%F A338223 a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).
%F A338223 a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).
%F A338223 a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.
%p A338223 g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
%p A338223       g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
%p A338223     end:
%p A338223 b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
%p A338223       g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A338223     end:
%p A338223 a:= n-> b(n, 2):
%p A338223 seq(a(n), n=2..37);  # _Alois P. Heinz_, Feb 10 2021
%t A338223 nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
%t A338223 nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
%t A338223 A015128[n_] := Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; a[n_] := (1/4) Sum[A015128[k] A015128[n - k], {k, 1, n - 1}]; Table[a[n], {n, 2, 37}]
%Y A338223 Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.
%K A338223 nonn
%O A338223 2,2
%A A338223 _Ilya Gutkovskiy_, Jan 30 2021