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%I #13 Sep 25 2022 15:01:32
%S 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 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 1,2,3,3,2,2,3,1,3,3,2,4,2,2,6,2,4,2,4
%N Number of partitions of n into distinct positive triangular numbers such that the number of parts is a triangular number.
%H Alois P. Heinz, <a href="/A357352/b357352.txt">Table of n, a(n) for n = 0..20000</a>
%e a(56) = 2 because we have [45,10,1] and [21,15,10,6,3,1].
%p b:= proc(n, i, t) option remember; (h-> `if`(n=0,
%p `if`(issqr(8*t+1), 1, 0), `if`(n>i*(i+1)*(i+2)/6, 0,
%p `if`(h>n, 0, b(n-h, i-1, t+1))+b(n, i-1, t))))(i*(i+1)/2)
%p end:
%p a:= n-> b(n, floor((sqrt(1+8*n)-1)/2), 0):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Sep 25 2022
%Y Cf. A000217, A024940, A045450, A178927, A357354.
%K nonn
%O 0,11
%A _Ilya Gutkovskiy_, Sep 25 2022