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A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.

%I A094040 #15 Jul 29 2018 10:32:27
%S A094040 1,1,1,1,3,3,1,6,14,12,1,10,40,75,55,1,15,90,275,429,273,1,21,175,770,
%T A094040 1911,2548,1428,1,28,308,1820,6370,13328,15504,7752,1,36,504,3822,
%U A094040 17640,51408,93024,95931,43263,1,45,780,7350,42840,162792,406980,648945,600875,246675
%N A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.
%C A094040 T(n,n-1) yields A001764; T(n,n-2) yields A026004.
%H A094040 Andrew Howroyd, <a href="/A094040/b094040.txt">Table of n, a(n) for n = 1..1275</a>
%H A094040 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.
%F A094040 T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).
%e A094040 Triangle begins:
%e A094040   1;
%e A094040   1,  1;
%e A094040   1,  3,   3;
%e A094040   1,  6,  14,  12;
%e A094040   1, 10,  40,  75,   55;
%e A094040   1, 15,  90, 275,  429,  273;
%e A094040   1, 21, 175, 770, 1911, 2548, 1428;
%e A094040   ...
%e A094040 T(3,1)=3 because the noncrossing forests on 3 vertices A,B,C and having one edge are (A, BC), (B, CA) and (C, AB).
%p A094040 T:=proc(n,k) if k<=n-1 then binomial(n,k+1)*binomial(n+2*k-1,k)/(n+k) else 0 fi end: seq(seq(T(n,k),k=0..n-1),n=1..11);
%t A094040 T[n_, k_] := Binomial[n, k+1] Binomial[n+2k-1, k]/(n+k);
%t A094040 Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Jul 29 2018 *)
%o A094040 (PARI)
%o A094040 T(n,k)=binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k);
%o A094040 for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ _Andrew Howroyd_, Nov 17 2017
%Y A094040 Cf. A001764, A026004, A045739, A094021.
%K A094040 nonn,tabl
%O A094040 1,5
%A A094040 _Emeric Deutsch_, May 31 2004