A094040 - OEIS

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

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OFFSET

1,5

COMMENTS

T(n,n-1) yields A001764; T(n,n-2) yields A026004.

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.

FORMULA

T(n, k)=binomial(n, k+1)*binomial(n+2k-1, k)/(n+k) (0<=k<=n-1).

EXAMPLE

Triangle begins:

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).

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);

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 *)

PROG

(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

CROSSREFS

Cf. A001764, A026004, A045739, A094021.

Sequence in context: A143389 A219218 A208524 * A039798 A193560 A278390

Adjacent sequences: A094037 A094038 A094039 * A094041 A094042 A094043

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 31 2004

STATUS

approved