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%I #7 Dec 13 2020 09:31:03
%S 1,0,1,0,1,1,1,1,1,2,2,2,1,2,3,2,3,2,3,3,5,2,4,4,4,5,3,5,5,7,4,5,6,5,
%T 8,6,6,7,8,6,7,9,8,8,9,8,10,10,8,11,10,9,10,11,11,12,14,8,13,14,13,11,
%U 13,14,16,15,10,16,16,15,15,16,16,16,21,12,18
%N Number of partitions of n into exactly five positive triangular numbers.
%H Alois P. Heinz, <a href="/A319815/b319815.txt">Table of n, a(n) for n = 5..10000</a>
%F a(n) = [x^n y^5] 1/Product_{j>=1} (1-y*x^A000217(j)).
%p h:= proc(n) option remember; `if`(n<1, 0,
%p `if`(issqr(8*n+1), n, h(n-1)))
%p end:
%p b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0), `if`(
%p k>n or i*k<n, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
%p end:
%p a:= n-> b(n, h(n), 5):
%p seq(a(n), n=5..120);
%t h[n_] := h[n] = If[n < 1, 0, If[IntegerQ@ Sqrt[8n + 1], n, h[n - 1]]];
%t b[n_, i_, k_] := b[n, i, k] = If[n==0, If[k==0, 1, 0], If[k > n || i k < n, 0, b[n, h[i - 1], k] + b[n - i, h[Min[n - i, i]], k - 1]]];
%t a[n_] := b[n, h[n], 5];
%t a /@ Range[5, 120] (* _Jean-François Alcover_, Dec 13 2020, after _Alois P. Heinz_ *)
%Y Column k=5 of A319797.
%Y Cf. A000217.
%K nonn
%O 5,10
%A _Alois P. Heinz_, Sep 28 2018
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