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A101371 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges and k leaves at level 1. 2
1, 0, 1, 2, 0, 1, 7, 4, 0, 1, 34, 14, 6, 0, 1, 171, 72, 21, 8, 0, 1, 905, 370, 114, 28, 10, 0, 1, 4952, 1995, 597, 160, 35, 12, 0, 1, 27802, 11064, 3278, 852, 210, 42, 14, 0, 1, 159254, 62774, 18420, 4762, 1135, 264, 49, 16, 0, 1, 927081, 362614, 105618, 27104, 6455, 1446, 322, 56, 18, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Row n has n+1 terms. Row sums give A001764. Column k=0 gives A023053.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.

P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.

M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

FORMULA

T(n, k) = Sum_{i=0..n-k} (-1)^i*((k+i+1)/(2n-k-i+1)) binomial(k+i, i) binomial(3n-2k-2i, n-k-i) for 0 <= k <= n.

G.f.: g/(1+z*g-t*z*g), where g = 1+z*g^3.

EXAMPLE

Triangle begins:

    1;

    0,  1;

    2,  0,  1;

    7,  4,  0, 1;

   34, 14,  6, 0, 1;

  171, 72, 21, 8, 0, 1;

  ...

MAPLE

T:=proc(n, k) if k<=n then sum((-1)^i*(k+i+1)*binomial(k+i, i)*binomial(3*n-2*k-2*i, n-k-i)/(2*n-k-i+1), i=0..n-k) else 0 fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

MATHEMATICA

t[n_, k_] := Sum[(-1)^i*(k + i + 1)/(2n - k - i + 1)*Binomial[k + i, i]* Binomial[3n - 2k - 2i, n - k - i], {i, 0, n - k}]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 21 2013, after Maple *)

PROG

(PARI) T(n, k) = sum(i=0, n-k, (-1)^i*(k+i+1)*binomial(k+i, i)*binomial(3*n-2*k-2*i, n-k-i)/(2*n-k-i+1)); \\ Andrew Howroyd, Nov 06 2017

CROSSREFS

Cf. A001764, A023053.

Sequence in context: A260693 A176129 A300130 * A154974 A291820 A078341

Adjacent sequences:  A101368 A101369 A101370 * A101372 A101373 A101374

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Jan 14 2005

STATUS

approved

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Last modified October 22 23:18 EDT 2018. Contains 316518 sequences. (Running on oeis4.)