A280952 Expansion of Product_{k>=0} 1/(1 - x^(5*k*(k+1)/2+1)).

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 56, 58, 60, 62, 64, 67, 68, 71, 74, 77, 79, 83, 85, 88, 91, 94, 96, 100

Offset

0,7

Comments

Number of partitions of n into centered pentagonal numbers (A005891).

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, Centered Pentagonal 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^(5*k*(k+1)/2+1)).

Example

a(12) = 3 because we have [6, 6], [6, 1, 1, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1].

Maple

h:= proc(n) option remember; `if`(n<0, 0, (t->
      `if`(((t+1)*5*t+2)/2>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))))(((i+1)*5*i+2)/2)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

Mathematica

nmax = 88; CoefficientList[Series[Product[1/(1 - x^(5 k (k + 1)/2 + 1)), {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A005891, A218379, A279221, A280950, A280951, A280953.

Sequence in context: A134119 A064661 A226982 * A235122 A131996 A090618

Adjacent sequences: A280949 A280950 A280951 * A280953 A280954 A280955

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 11 2017

Status

approved