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
Keywords
Examples
a(6) = 2 because we have [5, 1] and [1, 1, 1, 1, 1, 1].
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
Programs
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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]
Formula
G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).
Comments