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A241631 Number of compositions of n with exactly six descents.

%I A241631 #9 Nov 28 2023 09:12:34
%S A241631 13,139,927,4719,20122,75317,254805,794475,2315005,6369689,16681342,
%T A241631 41842970,101039601,235855414,534058830,1176497139,2527742988,
%U A241631 5308195367,10915439320,22015220687,43612884462,84970442998,162994589899,308155639555,574716486444
%N A241631 Number of compositions of n with exactly six descents.
%H A241631 Joerg Arndt and Alois P. Heinz, <a href="/A241631/b241631.txt">Table of n, a(n) for n = 18..1000</a>
%e A241631 a(18) = 13: [3,2,1,3,2,1,3,2,1], [2,1,3,2,1,3,2,1,2,1], [2,1,2,1,3,2,1,3,2,1], [3,2,1,3,2,1,2,1,2,1], [2,1,3,2,1,2,1,3,2,1], [3,2,1,2,1,2,1,3,2,1], [3,2,1,2,1,3,2,1,2,1], [2,1,2,1,2,1,3,2,1,2,1], [2,1,2,1,3,2,1,2,1,2,1], [2,1,3,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,2,1,3,2,1], [3,2,1,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,2,1,2,1,2,1].
%p A241631 b:= proc(n, i) option remember;
%p A241631       `if`(n=0, 1, convert(series(add(b(n-j, j)*
%p A241631       `if`(j<i, x, 1), j=1..n), x, 7), polynom))
%p A241631     end:
%p A241631 a:= n-> coeff(b(n, 0), x, 6):
%p A241631 seq(a(n), n=18..50);
%t A241631 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, j]*
%t A241631    If[j < i, x, 1], {j, 1, n}] // Expand];
%t A241631 a[n_] := Coefficient[b[n, 0], x, 6];
%t A241631 Table[a[n], {n, 18, 50}] (* _Jean-François Alcover_, Nov 28 2023, from Maple code *)
%Y A241631 Column k=6 of A238343 and of A238344.
%K A241631 nonn
%O A241631 18,1
%A A241631 _Joerg Arndt_ and _Alois P. Heinz_, Apr 26 2014