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A331919 Number of compositions (ordered partitions) of n into distinct tetrahedral numbers.

%I A331919 #20 Jun 20 2020 10:13:56
%S A331919 1,1,0,0,1,2,0,0,0,0,1,2,0,0,2,6,0,0,0,0,1,2,0,0,2,6,0,0,0,0,2,6,0,0,
%T A331919 6,25,2,0,0,2,6,0,0,0,0,2,6,0,0,6,24,0,0,0,0,2,7,2,0,6,26,6,0,0,0,6,
%U A331919 26,6,0,24,126,24,0,0,0,0,2,6,0,0,6,24,0,0,1,2,6,24,2,6,24
%N A331919 Number of compositions (ordered partitions) of n into distinct tetrahedral numbers.
%H A331919 Robert Israel, <a href="/A331919/b331919.txt">Table of n, a(n) for n = 0..10000</a>
%H A331919 <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%e A331919 a(15) = 6 because we have [10, 4, 1], [10, 1, 4], [4, 10, 1], [4, 1, 10], [1, 10, 4] and [1, 4, 10].
%p A331919 N:= 200: # for a(0)..a(N)
%p A331919 G:= mul(1+t*x^(i*(i+1)*(i+2)/6), i=1..floor((6*N)^(1/3))):
%p A331919 F:= proc(n) local R, k, v;
%p A331919   R:= coeff(G, x, n);
%p A331919   add(k!*coeff(R, t, k), k=1..degree(R, t))
%p A331919 end proc:
%p A331919 F(0):= 1:
%p A331919 map(F, [$0..N]); # _Robert Israel_, Feb 03 2020
%t A331919 M = 100;
%t A331919 G = Product[1 + t x^(i(i+1)(i+2)/6), {i, 1, Floor[(6M)^(1/3)]}];
%t A331919 F[n_] := Module[{R, k, v}, R = Coefficient[G, x, n]; Sum[k! Coefficient[R, t, k], {k, 1, Exponent[R, t]}]];
%t A331919 F[0] = 1;
%t A331919 F /@ Range[0, M] (* _Jean-François Alcover_, Jun 20 2020, after _Robert Israel_ *)
%Y A331919 Cf. A000292, A023361, A068980, A279278, A282582, A298857, A331843, A331984.
%K A331919 nonn,look
%O A331919 0,6
%A A331919 _Ilya Gutkovskiy_, Feb 01 2020