login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A108410 Triangle T(n,k) read by rows: number of 12312-avoiding matchings on [2n] with exactly k crossings (n >= 1, 0 <= k <= n-1). 3
1, 2, 1, 5, 5, 2, 14, 21, 15, 5, 42, 84, 84, 49, 14, 132, 330, 420, 336, 168, 42, 429, 1287, 1980, 1980, 1350, 594, 132, 1430, 5005, 9009, 10725, 9075, 5445, 2145, 429, 4862, 19448, 40040, 55055, 55055, 40898, 22022, 7865, 1430, 16796, 75582 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Alexander Burstein, Megan Martinez, Pattern classes equinumerous to the class of ternary forests, Permutation Patterns Virtual Workshop, Howard University (2020).

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, arXiv:math/0504342 [math.CO], 2005.

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 2.2.

D. S. Hough, Descents in noncrossing trees, Electronic J. Combinatorics 10 (2003), #N13, Theorem 2.2. [Ira M. Gessel, May 10 2010]

FORMULA

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

T(n,k) = C(n-1+k,n-1)*C(2*n-k,n+1)/n, (0 <= k <= n-1). [Chen et al.] - Emeric Deutsch, Dec 19 2006

O.g.f. equals the series reversion with respect to x of x*(1 + x*(1 - t))/(1 + x)^3. If R(n,t) is the n-th row polynomial of this triangle then R(n, 1+t) is the n-th row polynomial of A089434. - Peter Bala, Jul 15 2012

EXAMPLE

Triangle begins

     1;

     2,     1;

     5,     5,     2;

    14,    21,    15,     5;

    42,    84,    84,    49,    14;

   132,   330,   420,   336,   168,    42;

   429,  1287,  1980,  1980,  1350,   594,   132;

  1430,  5005,  9009, 10725,  9075,  5445,  2145,  429;

  4862, 19448, 40040, 55055, 55055, 40898, 22022, 7865, 1430;

MAPLE

T:=(n, k)->binomial(n-1+k, n-1)*binomial(2*n-k, n+1)/n: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 19 2006

MATHEMATICA

T[n_, k_] := Binomial[n + k - 1, n - 1]*Binomial[2*n - k, n + 1]/n;

Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Emeric Deutsch *)

PROG

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

CROSSREFS

Left-hand columns include A000108 and A002054. Right-hand columns include A000108 and A007851+1. Row sums are A001764. A089434.

Sequence in context: A046757 A248905 A118244 * A058116 A058118 A332632

Adjacent sequences:  A108407 A108408 A108409 * A108411 A108412 A108413

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Jun 03 2005

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 30 05:46 EST 2021. Contains 349419 sequences. (Running on oeis4.)