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

1, 4, 12, 30, 68, 144, 289, 556, 1034, 1868, 3292, 5678, 9608, 15984, 26188, 42314, 67509, 106460, 166090, 256552, 392628, 595696, 896484, 1338894, 1985298, 2923840, 4278448, 6222518, 8997544, 12938368, 18507297, 26340040, 37307326, 52597320, 73825504, 103180702

Offset: 2

Author

Ilya Gutkovskiy, Jan 30 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A001934, A002448, A014968, A015128, A048574, A063725, A327380.

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, 2):
seq(a(n), n=2..37);  # Alois P. Heinz, Feb 10 2021

Mathematica

nmax = 37; CoefficientList[Series[(1/EllipticTheta[4, 0, x] - 1)^2/4, {x, 0, nmax}], x] // Drop[#, 2] &
nmax = 37; CoefficientList[Series[(1/4) (-1 + Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}])^2, {x, 0, nmax}], x] // Drop[#, 2] &
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}]

Formula

G.f.: (1/4) * (-1 + Product_{k>=1} (1 + x^k) / (1 - x^k))^2.

a(n) = Sum_{k=0..n} A014968(k) * A014968(n-k).

a(n) = (1/4) * Sum_{k=1..n-1} A015128(k) * A015128(n-k).

a(n) = (A001934(n) - 2 * A015128(n)) / 4 for n > 0.