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A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.

%I A331900 #13 Nov 17 2020 05:40:21
%S A331900 1,1,1,1,7,1,3,13,3,55,201,159,865,1803,7093,43431,14253,22903,130851,
%T A331900 120763,1099693,4527293,4976767,7516897,14349685,72866239,81946383,
%U A331900 167841291,897853735,455799253,946267825,5054280915,3941268001,17066300985,49111862599
%N A331900 Number of compositions (ordered partitions) of the n-th triangular number into distinct triangular numbers.
%H A331900 Alois P. Heinz, <a href="/A331900/b331900.txt">Table of n, a(n) for n = 0..300</a>
%H A331900 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F A331900 a(n) = A331843(A000217(n)).
%e A331900 a(6) = 3 because we have [21], [15, 6] and [6, 15].
%p A331900 b:= proc(n, i, p) option remember; (t->
%p A331900       `if`(t*(i+2)/3<n, 0, `if`(n=0, p!, b(n, i-1, p)+
%p A331900       `if`(t>n, 0, b(n-t, i-1, p+1)))))((i*(i+1)/2))
%p A331900     end:
%p A331900 a:= n-> b(n*(n+1)/2, n, 0):
%p A331900 seq(a(n), n=0..37);  # _Alois P. Heinz_, Jan 31 2020
%t A331900 b[n_, i_, p_] := b[n, i, p] = With[{t = i(i+1)/2}, If[t(i+2)/3 < n, 0, If[n == 0, p!, b[n, i-1, p] + If[t > n, 0, b[n-t, i-1, p+1]]]]];
%t A331900 a[n_] := b[n(n+1)/2, n, 0];
%t A331900 a /@ Range[0, 37] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A331900 Cf. A000217, A072964, A126683, A196010, A224677, A288126, A298858, A307614, A331843, A331884.
%K A331900 nonn
%O A331900 0,5
%A A331900 _Ilya Gutkovskiy_, Jan 31 2020
%E A331900 More terms from _Alois P. Heinz_, Jan 31 2020