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

1, 8, 40, 156, 520, 1552, 4262, 10960, 26716, 62276, 139744, 303412, 640001, 1315832, 2644004, 5204044, 10052182, 19086348, 35672516, 65708116, 119409576, 214289116, 380068582, 666723748, 1157550524, 1990230968, 3390558072, 5726064688, 9590759624, 15938198484, 26289242026

Offset: 4

Author

Ilya Gutkovskiy, Feb 10 2021

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 4..10000

Crossrefs

Cf. A002448, A004405, A014968, A015128, A063730, A284286, A327382, A338223, A341222, A341364, A341366, A341367, A341368, A341369, A341370.

Programs

Maple

g:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i=1, 0,
      g(n, i-1))+add(2*g(n-i*j, i-1), j=`if`(i=1, n, 1)..n/i))
    end:
b:= proc(n, k) option remember; `if`(k=0, 1, `if`(k=1, `if`(n=0, 0,
      g(n$2)), (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..34);  # Alois P. Heinz, Feb 10 2021

Mathematica

nmax = 34; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^4/16, {x, 0, nmax}], x] // Drop[#, 4] &
nmax = 34; CoefficientList[Series[(1/16) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (1/16) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^4.

a(n) ~ A284286(n)/16. - Vaclav Kotesovec, Feb 20 2021