A279951 Expansion of Product_{k>=1} 1/(1 - x^((k*(k+1)/2)^2)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 10, 12, 12, 12, 12, 12, 12, 12, 12, 12, 15, 15, 15, 15, 15, 15, 15, 15, 15, 18, 18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21, 21, 21, 21, 21, 21, 24, 25, 25, 25, 25, 25, 25

Offset

0,10

Comments

Number of partitions of n into nonzero squared triangular numbers (A000537).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Wikipedia, Squared triangular number

Index entries for related partition-counting sequences

Formula

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

Example

a(10) = 2 because we have [9, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Mathematica

nmax = 105; CoefficientList[Series[Product[1/(1 - x^((k*(k + 1)/2)^2)), {k, 1, Floor[Sqrt[1 + 8*Sqrt[nmax]]/2] + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 18 2026 *)

Crossrefs

Cf. A000537, A007294, A068980.

Sequence in context: A133879 A343609 A054898 * A279224 A167383 A167661

Adjacent sequences: A279948 A279949 A279950 * A279952 A279953 A279954

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 23 2016

Status

approved