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