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A322340 Number of compositions (ordered partitions) of n into square pyramidal numbers (A000330).

%I A322340 #19 Feb 16 2025 08:33:57
%S A322340 1,1,1,1,1,2,3,4,5,6,8,11,15,20,27,36,48,64,85,114,153,205,274,365,
%T A322340 487,651,871,1165,1557,2080,2780,3716,4967,6639,8873,11860,15853,
%U A322340 21189,28320,37850,50589,67618,90379,120799,161456,215797,288430,385512,515269,688699
%N A322340 Number of compositions (ordered partitions) of n into square pyramidal numbers (A000330).
%H A322340 Alois P. Heinz, <a href="/A322340/b322340.txt">Table of n, a(n) for n = 0..7939</a>
%H A322340 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H A322340 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A322340 G.f.: 1/(1 - Sum_{k>=1} x^(k*(k+1)*(2*k+1)/6)).
%p A322340 h:= proc(n) option remember; `if`(n<1, 0, (t->
%p A322340       `if`(t*(t+1)*(2*t+1)/6>n, t-1, t))(1+h(n-1)))
%p A322340     end:
%p A322340 a:= proc(n) option remember; `if`(n=0, 1,
%p A322340       add(a(n-i*(i+1)*(2*i+1)/6), i=1..h(n)))
%p A322340     end:
%p A322340 seq(a(n), n=0..60);  # _Alois P. Heinz_, Dec 28 2018
%t A322340 nmax = 49; CoefficientList[Series[1/(1 - Sum[x^(k (k + 1) (2 k + 1)/6), {k, 1, nmax}]), {x, 0, nmax}], x]
%Y A322340 Cf. A000330, A006456, A279220, A282582, A298246.
%K A322340 nonn
%O A322340 0,6
%A A322340 _Ilya Gutkovskiy_, Dec 26 2018