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A241628 Number of compositions of n with exactly three descents.

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