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

2,2

Links

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

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.

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}]

Crossrefs

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

Sequence in context: A051172 A032192 A212587 * A118425 A097809 A272144

Adjacent sequences: A338220 A338221 A338222 * A338224 A338225 A338226

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 30 2021

Status

approved