A280953 Expansion of Product_{k>=0} 1/(1 - x^(3*k*(k+1)+1)).

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 20, 22, 23, 24, 25, 25, 27, 27, 29, 30, 31, 32, 32, 34, 34, 36, 37, 38, 39, 40, 43, 44, 46, 47, 48, 50, 51, 54, 55, 57, 58, 59

Offset

0,8

Comments

Number of partitions of n into centered hexagonal numbers (A003215).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000

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, Hex 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)+1)).

Example

a(14) = 3 because we have [7, 7], [7, 1, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Maple

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(3*t*(t+1)+1>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(3*i*(i+1)+1)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

Mathematica

nmax = 86; CoefficientList[Series[Product[1/(1 - x^(3 k (k + 1) + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A003215, A278949, A279222, A280950, A280951, A280952.

Sequence in context: A296237 A074802 A071804 * A111894 A279012 A292027

Adjacent sequences: A280950 A280951 A280952 * A280954 A280955 A280956

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 11 2017

Status

approved