A241627 - OEIS

%I #10 Aug 27 2021 03:17:13

%S 2,8,29,81,205,469,1013,2059,4021,7558,13780,24440,42358,71867,119715,

%T 196084,316362,503410,791043,1228636,1888003,2872541,4330299,6471778,

%U 9594556,14116745,20622825,29925512,43149302,61843197,88130983,124912824,176132457

%N Number of compositions of n with exactly two descents.

%H Joerg Arndt and Alois P. Heinz, <a href="/A241627/b241627.txt">Table of n, a(n) for n = 6..1000</a>

%e a(6) = 2: [3,2,1], [2,1,2,1].

%e a(7) = 8: [4,2,1], [3,2,1,1], [3,1,2,1], [1,3,2,1], [2,1,3,1], [1,2,1,2,1], [2,1,1,2,1], [2,1,2,1,1].

%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, 3), polynom))

%p     end:

%p a:= n-> coeff(b(n, 0), x, 2):

%p seq(a(n), n=6..50);

%t k = 2;

%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[6, 50] (* _Jean-François Alcover_, Aug 27 2021, after Maple code *)

%Y Column k=2 of A238343 and of A238344.

%K nonn

%O 6,1

%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 26 2014