|
%I #11 Dec 28 2018 19:32:27
%S 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,
%T 13,14,15,17,17,17,18,19,21,21,22,23,25,27,27,28,29,31,33,33,34,35,37,
%U 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
%N Expansion of Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).
%C Number of partitions of n into nonzero square pyramidal numbers (A000330).
%H Alois P. Heinz, <a href="/A279220/b279220.txt">Table of n, a(n) for n = 0..20000</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Product_{k>=1} 1/(1 - x^(k*(k+1)*(2*k+1)/6)).
%e a(6) = 2 because we have [5, 1] and [1, 1, 1, 1, 1, 1].
%p h:= proc(n) option remember; `if`(n<1, 0, (t->
%p `if`(t*(t+1)*(2*t+1)/6>n, t-1, t))(1+h(n-1)))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(i+1)*(2*i+1)/6)))
%p end:
%p a:= n-> b(n, h(n)):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Dec 28 2018
%t nmax=90; CoefficientList[Series[Product[1/(1 - x^(k (k + 1) (2 k + 1)/6)), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000330, A068980, A279221, A279222, A279223, A279224.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, Dec 08 2016
|
