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

%I A357352 #16 Apr 23 2025 13:30:20
%S A357352 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,
%T A357352 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,
%U A357352 1,2,3,3,2,2,3,1,3,3,2,4,2,2,6,2,4,2,4
%N A357352 Number of partitions of n into distinct positive triangular numbers such that the number of parts is a triangular number.
%H A357352 Alois P. Heinz, <a href="/A357352/b357352.txt">Table of n, a(n) for n = 0..20000</a>
%e A357352 a(56) = 2 because we have [45,10,1] and [21,15,10,6,3,1].
%p A357352 b:= proc(n, i, t) option remember; (h-> `if`(n=0,
%p A357352      `if`(issqr(8*t+1), 1, 0), `if`(n>i*(i+1)*(i+2)/6, 0,
%p A357352      `if`(h>n, 0, b(n-h, i-1, t+1))+b(n, i-1, t))))(i*(i+1)/2)
%p A357352     end:
%p A357352 a:= n-> b(n, floor((sqrt(1+8*n)-1)/2), 0):
%p A357352 seq(a(n), n=0..100);  # _Alois P. Heinz_, Sep 25 2022
%t A357352 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]]]];
%t A357352 a[n_] := b[n, Floor[(Sqrt[8n+1]-1)/2], 0];
%t A357352 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Apr 22 2025, after _Alois P. Heinz_ *)
%Y A357352 Cf. A000217, A024940, A045450, A178927, A357354.
%K A357352 nonn
%O A357352 0,11
%A A357352 _Ilya Gutkovskiy_, Sep 25 2022