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%I #10 Aug 27 2021 03:17:02
%S 3,18,78,264,786,2097,5179,11998,26400,55593,112814,221639,423318,
%T 788518,1436302,2564135,4494967,7750068,13160903,22039386,36434095,
%U 59514365,96139570,153699716,243345157,381779187,593848668,916277405,1403004758,2132797015
%N Number of compositions of n with exactly three descents.
%H Joerg Arndt and Alois P. Heinz, <a href="/A241628/b241628.txt">Table of n, a(n) for n = 9..1000</a>
%e a(9) = 3: [2,1,3,2,1], [3,2,1,2,1], [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, 4), polynom))
%p end:
%p a:= n-> coeff(b(n, 0), x, 3):
%p seq(a(n), n=9..50);
%t k = 3;
%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[9, 50] (* _Jean-François Alcover_, Aug 27 2021, after Maple code *)
%Y Column k=3 of A238343 and of A238344.
%K nonn
%O 9,1
%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 26 2014
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