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

0,11

Links

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A220400 A285108 A030216 * A339376 A362425 A159459

Adjacent sequences: A357349 A357350 A357351 * A357353 A357354 A357355

Keyword

nonn

Author

Ilya Gutkovskiy, Sep 25 2022

Status

approved