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A319814 Number of partitions of n into exactly four positive triangular numbers.

%I A319814 #10 Dec 13 2020 09:30:56
%S A319814 1,0,1,0,1,1,1,1,1,2,1,2,1,1,3,2,2,1,3,2,4,2,2,3,3,3,2,4,3,5,3,2,5,4,
%T A319814 4,3,5,4,4,5,4,5,5,4,6,5,5,6,5,5,6,7,3,5,9,5,7,5,8,7,7,4,7,9,7,8,5,7,
%U A319814 8,10,6,6,10,7,10,7,8,9,8,8,7,13,7,10,11
%N A319814 Number of partitions of n into exactly four positive triangular numbers.
%H A319814 Alois P. Heinz, <a href="/A319814/b319814.txt">Table of n, a(n) for n = 4..10000</a>
%F A319814 a(n) = [x^n y^4] 1/Product_{j>=1} (1-y*x^A000217(j)).
%p A319814 h:= proc(n) option remember; `if`(n<1, 0,
%p A319814       `if`(issqr(8*n+1), n, h(n-1)))
%p A319814     end:
%p A319814 b:= proc(n, i, k) option remember; `if`(n=0, `if`(k=0, 1, 0), `if`(
%p A319814       k>n or i*k<n, 0, b(n, h(i-1), k)+b(n-i, h(min(n-i, i)), k-1)))
%p A319814     end:
%p A319814 a:= n-> b(n, h(n), 4):
%p A319814 seq(a(n), n=4..120);
%t A319814 h[n_] := h[n] = If[n<1, 0, If[IntegerQ@Sqrt[8n + 1], n, h[n - 1]]];
%t A319814 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 A319814 a[n_] := b[n, h[n], 4];
%t A319814 a /@ Range[4, 120] (* _Jean-François Alcover_, Dec 13 2020, after _Alois P. Heinz_ *)
%Y A319814 Column k=4 of A319797.
%Y A319814 Cf. A000217.
%K A319814 nonn
%O A319814 4,10
%A A319814 _Alois P. Heinz_, Sep 28 2018