A357352 Number of partitions of n into distinct positive triangular numbers such that the number of parts is a triangular number.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 0, 1, 1, 1, 0, 3, 0, 1, 1, 2, 0, 1, 1, 1, 3, 0, 2, 1, 1, 1, 1, 2, 2, 2, 1, 0, 3, 1, 0, 4, 1, 2, 2, 2, 1, 2, 2, 1, 3, 1, 3, 2, 1, 3, 3, 1, 2, 3, 3, 2, 2, 3, 1, 3, 3, 2, 4, 2, 2, 6, 2, 4, 2, 4

Offset: 0

Author

Ilya Gutkovskiy, Sep 25 2022

Keywords

• nonn

Examples

			a(56) = 2 because we have [45,10,1] and [21,15,10,6,3,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A000217, A024940, A045450, A178927, A357354.

Programs

Maple

b:= proc(n, i, t) option remember; (h-> `if`(n=0,
     `if`(issqr(8*t+1), 1, 0), `if`(n>i*(i+1)*(i+2)/6, 0,
     `if`(h>n, 0, b(n-h, i-1, t+1))+b(n, i-1, t))))(i*(i+1)/2)
    end:
a:= n-> b(n, floor((sqrt(1+8*n)-1)/2), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 25 2022

Mathematica

b[n_, i_, t_] := b[n, i, t] = With[{h = i(i+1)/2}, If[n == 0, If[IntegerQ@ Sqrt[8t+1], 1, 0], If[n > i(i+1)(i+2)/6, 0, If[h > n, 0, b[n-h, i-1, t+1]] + b[n, i-1, t]]]];
a[n_] := b[n, Floor[(Sqrt[8n+1]-1)/2], 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 22 2025, after Alois P. Heinz *)