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 A319815 Number of partitions of n into exactly five positive triangular numbers. 5

%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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Last modified October 4 22:25 EDT 2023. Contains 365888 sequences. (Running on oeis4.)