A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges.
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, 1911, 2548, 1428, 1, 28, 308, 1820, 6370, 13328, 15504, 7752, 1, 36, 504, 3822, 17640, 51408, 93024, 95931, 43263, 1, 45, 780, 7350, 42840, 162792, 406980, 648945, 600875, 246675
Offset: 1
Examples
Triangle begins: 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, 1911, 2548, 1428; ... 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).
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
- P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
Programs
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Maple
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);
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Mathematica
T[n_, k_] := Binomial[n, k+1] Binomial[n+2k-1, k]/(n+k); Table[T[n, k], {n, 1, 11}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 29 2018 *) -
PARI
T(n,k)=binomial(n, k+1)*binomial(n+2*k-1, k)/(n+k); for(n=1, 10, for(k=0, n-1, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017
Formula
T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).
Comments