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A241629 Number of compositions of n with exactly four descents.

%I A241629 #9 Aug 27 2021 03:16:52
%S A241629 5,37,189,751,2558,7762,21577,55867,136478,317416,707817,1521755,
%T A241629 3168262,6410970,12646075,24378923,46028497,85269218,155238256,
%U A241629 278130152,490983234,854910432,1469679627,2496560526,4193815403,6971437570,11474937414,18712755604
%N A241629 Number of compositions of n with exactly four descents.
%H A241629 Joerg Arndt and Alois P. Heinz, <a href="/A241629/b241629.txt">Table of n, a(n) for n = 12..1000</a>
%e A241629 a(12) = 5: [3,2,1,3,2,1], [2,1,2,1,3,2,1], [2,1,3,2,1,2,1], [3,2,1,2,1,2,1], [2,1,2,1,2,1,2,1].
%p A241629 b:= proc(n, i) option remember;
%p A241629       `if`(n=0, 1, convert(series(add(b(n-j, j)*
%p A241629       `if`(j<i, x, 1), j=1..n), x, 5), polynom))
%p A241629     end:
%p A241629 a:= n-> coeff(b(n, 0), x, 4):
%p A241629 seq(a(n), n=12..50);
%t A241629 k = 4;
%t A241629 b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
%t A241629      If[j < i, x, 1], {j, n}] + O[x]^(k + 1)];
%t A241629 a[n_] := SeriesCoefficient[b[n, 0], {x, 0, k}];
%t A241629 a /@ Range[12, 50] (* _Jean-François Alcover_, Aug 27 2021, after Maple code *)
%Y A241629 Column k=4 of A238343 and of A238344.
%K A241629 nonn
%O A241629 12,1
%A A241629 _Joerg Arndt_ and _Alois P. Heinz_, Apr 26 2014