A280950 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+1)).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9, 11, 11, 12, 13, 15, 15, 16, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 37, 40, 43, 45, 48, 51, 54, 56, 60, 63, 67, 70, 76, 80, 84, 87, 93, 97, 102, 106, 113, 118, 125, 130, 138, 143, 151, 157, 166, 172, 181, 189, 200, 207, 217, 225, 237, 245, 257, 267, 280

Offset

0,5

Comments

Number of partitions of n into centered triangular numbers (A005448).

Links

Robert Israel, 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]

Eric Weisstein's World of Mathematics, Centered Triangular Number

Index entries for sequences related to centered polygonal numbers

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=0} 1/(1 - x^(3*k*(k+1)/2+1)).

Example

a(8) = 3 because we have [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].

Maple

N:= 100:
kmax:= floor((sqrt(24*N-15)-3)/6):
S:= series(mul(1/(1-x^(3*k*(k+1)/2+1)), k=0..kmax), x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, Jan 25 2017

Mathematica

nmax = 78; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A005448, A007294, A068980, A280951, A280952, A280953.

Sequence in context: A185322 A324918 A068980 * A279135 A053266 A112217

Adjacent sequences: A280947 A280948 A280949 * A280951 A280952 A280953

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 11 2017

Status

approved