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%I #9 Aug 27 2021 03:16:40
%S 8,73,429,1945,7465,25317,78068,222963,597644,1518370,3683950,8586866,
%T 19320391,42127208,89307361,184578961,372786467,737212357,1429992711,
%U 2724852920,5107392644,9427895421,17157142969,30810379849,54643242181,95783964110,166059755226
%N Number of compositions of n with exactly five descents.
%H Joerg Arndt and Alois P. Heinz, <a href="/A241630/b241630.txt">Table of n, a(n) for n = 15..1000</a>
%e a(15) = 8: [3,2,1,2,1,3,2,1], [3,2,1,3,2,1,2,1], [2,1,3,2,1,3,2,1], [2,1,2,1,3,2,1,2,1], [2,1,3,2,1,2,1,2,1], [3,2,1,2,1,2,1,2,1], [2,1,2,1,2,1,3,2,1], [2,1,2,1,2,1,2,1,2,1].
%p b:= proc(n, i) option remember;
%p `if`(n=0, 1, convert(series(add(b(n-j, j)*
%p `if`(j<i, x, 1), j=1..n), x, 6), polynom))
%p end:
%p a:= n-> coeff(b(n, 0), x, 5):
%p seq(a(n), n=15..50);
%t k = 5;
%t b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n - j, j]*
%t If[j < i, x, 1], {j, n}] + O[x]^(k + 1)];
%t a[n_] := SeriesCoefficient[b[n, 0], {x, 0, k}];
%t a /@ Range[15, 50] (* _Jean-François Alcover_, Aug 27 2021, after Maple code *)
%Y Column k=5 of A238343 and of A238344.
%K nonn
%O 15,1
%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 26 2014
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