A279220 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7, 7, 7, 7, 8, 9, 9, 9, 10, 11, 13, 13, 13, 14, 15, 17, 17, 17, 18, 19, 21, 21, 22, 23, 25, 27, 27, 28, 29, 31, 33, 33, 34, 35, 37, 40, 41, 42, 44, 46, 50, 51, 52, 54, 56, 60, 61, 62, 64, 67, 72, 73, 75, 77, 81, 86, 87, 89, 91, 95, 100, 101, 103, 106, 111, 117, 119, 121, 125, 130, 137

Offset

0,6

Comments

Number of partitions of n into nonzero square pyramidal numbers (A000330).

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, Square Pyramidal Number

Index to sequences related to pyramidal numbers

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).

Example

a(6) = 2 because we have [5, 1] and [1, 1, 1, 1, 1, 1].

Maple

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(t+1)*(2*t+1)/6>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(i+1)*(2*i+1)/6)))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

Mathematica

nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (2 k + 1)/6)), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000330, A068980, A279221, A279222, A279223, A279224.

Sequence in context: A005857 A025809 A204591 * A114575 A131849 A362344

Adjacent sequences: A279217 A279218 A279219 * A279221 A279222 A279223

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 08 2016

Status

approved