A301334 a(n) = [x^n] 1/(1 + n*(1 - theta_2(sqrt(x))/(2*x^(1/8)))), where theta_2() is the Jacobi theta function.

1, 1, 4, 30, 288, 3500, 51882, 908705, 18376192, 421518897, 10815546010, 306954846231, 9547629128208, 322979502072591, 11805623386524688, 463679308850798265, 19474458473055138816, 870962008703995217038, 41324081662873427484240, 2073203796753598883831150, 109655938011610286565760400

Offset: 0

Author

Ilya Gutkovskiy, Mar 18 2018

Keywords

• nonn

Comments

Number of compositions (ordered partitions) of n into triangular numbers of n kinds.

Links

• Index entries for sequences related to compositions
• Index to sequences related to polygonal numbers

Crossrefs

Cf. A000217, A023361, A301335.

Programs

Mathematica

Table[SeriesCoefficient[1/(1 + n (1 - EllipticTheta[2, 0, Sqrt[x]]/(2 x^(1/8)))), {x, 0, n}], {n, 0, 20}]
Table[SeriesCoefficient[1/(1 - n Sum[x^(k (k + 1)/2), {k, 1, n}]), {x, 0, n}], {n, 0, 20}]

Formula

a(n) = [x^n] 1/(1 - n*Sum_{k>=1} x^(k*(k+1)/2)).

a(n) ~ n^n * (1 + 1/n - 3/(2*n^2) - 13/(3*n^3) + 181/(24*n^4) + 2251/(120*n^5) - 34949/(720*n^6) - 221539/(2520*n^7) + 13489169/(40320*n^8) + ...). - Vaclav Kotesovec, Mar 19 2018