A290792 Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 18, 18, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 27, 27, 28, 28, 30, 30, 32, 32, 34, 34, 36, 36, 38, 38, 40, 40, 42, 42, 45, 45, 47, 47, 49, 49, 52, 52, 54, 54, 57

Offset: 0

Author

Ilya Gutkovskiy, Aug 15 2017

Keywords

• nonn

Comments

Number of partitions of n into nonzero 4-dimensional pyramidal numbers (A002415).

Examples

			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].
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Index to sequences related to pyramidal numbers
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000330, A001156, A002415, A279220, A290573.

Programs

Maple

N:= 100: # for a(0)..a(N)
P:= 1:
for k from 1 do
  e:= k*(k+1)^2*(k+2)/12;
  if e > N then break fi;
  P:= P/(1-x^e);
od:
S:= series(P,x,N+1):
[seq](coeff(S,x,n),n=0..N); # Robert Israel, Aug 28 2019

Mathematica

nmax = 90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1)^2 (k + 2)/12)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)^2*(k+2)/12)).