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 A094040 Triangle read by rows: T(n,k) is the number of noncrossing forests with n vertices and k edges. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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;   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

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Last modified May 11 19:29 EDT 2021. Contains 343808 sequences. (Running on oeis4.)