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

0,3

Comments

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

Links

Table of n, a(n) for n=0..20.

Index entries for sequences related to compositions

Index to sequences related to polygonal numbers

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

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}]

Crossrefs

Cf. A000217, A023361, A301335.

Sequence in context: A379279 A052631 A368893 * A167139 A347994 A240958

Adjacent sequences: A301331 A301332 A301333 * A301335 A301336 A301337

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 18 2018

Status

approved