A072964 Number of partitions of n-th triangular number n(n+1)/2 (A000217(n)) into triangular parts.

1, 1, 2, 4, 7, 15, 32, 66, 141, 295, 619, 1286, 2654, 5460, 11066, 22357, 44962, 89258, 176459, 347103, 675846, 1309903, 2525893, 4830943, 9196093, 17418788, 32772432, 61375543, 114401182, 212026732, 391231769, 718710706, 1313781686

Offset: 0

Author

Paul D. Hanna, Aug 13 2002

Keywords

• nonn

Comments

What is limit_{n->inf} a(n)^(1/n)? [This limit is equal to 1. - Vaclav Kotesovec, May 21 2018]

Links

• T. D. Noe, Table of n, a(n) for n=0..100

Crossrefs

Cf. A007294, A073420, A114738.

Programs

Mathematica

c = CoefficientList[ Series[1/Product[1 - x^(i(i + 1)/2), {i, 1, 50}], {x, 0, 565}], x]; c[[Range[33]*(Range[33] + 1)/2 + 1]] (* Robert G. Wilson v *)

Formula

a(n) = A007294[n(n+1)/2] = coefficient of x^[n(n+1)/2] in the expansion of product_{k=1..inf} 1/(1 - x^(k(k+1)/2)).

a(n) = A007294(A000217(n)).

a(n) ~ exp(3*Pi^(1/3) * Zeta(3/2)^(2/3) * (n*(n+1))^(1/3) / 2^(4/3)) * Zeta(3/2) / (4*Pi*sqrt(3)*n^3). - Vaclav Kotesovec, May 21 2018

Extensions

Entry revised by N. J. A. Sloane, Jan 28 2007