%I #13 Jan 10 2023 07:19:26
%S 1,1,2,2,4,6,10,17,27,59,116,224,427,839,1616,3110,5968,11337,21560,
%T 41002,77058,144395,270041,500683,926185,1706838,3126023,5706968,
%U 10379752,18782112,33868690,60857056,108867121,194086938,344821828,610237587,1076401704
%N Number of partitions of the n-th triangular number n*(n+1)/2 into a triangular number of triangular parts.
%H Alois P. Heinz, <a href="/A344707/b344707.txt">Table of n, a(n) for n = 0..141</a>
%e a(5) = 6: [15], [10,1,1,1,1,1], [6,6,3], [6,3,3,1,1,1], [6,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1].
%p t:= n-> issqr(8*n+1):
%p h:= proc(n) option remember; `if`(t(n), n, h(n-1)) end:
%p b:= proc(n, i, c) option remember; `if`(n=0 or i=1, `if`(
%p t(c+n), 1, 0), b(n-i, h(min(n-i, i)), c+1)+b(n, h(i-1), c))
%p end:
%p a:= n-> b(n*(n+1)/2$2, 0):
%p seq(a(n), n=0..40);
%t t[n_] := IntegerQ@Sqrt[8n+1];
%t h[n_] := h[n] = If[t[n], n, h[n-1]];
%t b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1, If[t[c+n], 1, 0], b[n-i, h[Min[n-i, i]], c+1] + b[n, h[i-1], c]];
%t a[n_] := b[n(n+1)/2, n(n+1)/2, 0];
%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jan 10 2023, after _Alois P. Heinz_ *)
%Y Cf. A000217.
%K nonn
%O 0,3
%A _Alois P. Heinz_, May 26 2021