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A006150 Number of Dyck paths.
(Formerly M4013)
4
1, 1, 5, 55, 1001, 26026, 884884, 37119160, 1844536720, 105408179176, 6774025632340, 481155055944150, 37259723952950625, 3111129272480118750, 277587585343361452500, 26268551497229678505000, 2620002484114994890890000, 273961129317241857069150000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the determinant of the 4 X 4 Hankel matrix [a_0, a_1, a_2, a_3 ; a_1, a_2, a_3, a_4 ; a_2, a_3, a_4, a_5 ; a_3, a_4, a_5, a_6] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007

REFERENCES

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).

M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique. Ph.D Dissertation, Université Bordeaux I, 1983.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..431

M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique,  Ph.D Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy)

Nicholas M. Katz, A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers, 2015.

FORMULA

a(n) = Det[Table[binomial[i+3, j-i+4], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005

From Vaclav Kotesovec, Mar 20 2014: (Start)

Recurrence: (n+4)*(n+5)*(n+6)*(n+7)*a(n) = 16*(2*n-1)*(2*n+1)*(2*n+3)*(2*n+5)*a(n-1).

a(n) = 3628800 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)!).

a(n) ~ 14863564800 * 256^n / (Pi^2 * n^18).

(End)

MAPLE

with(LinearAlgebra):

ctln:= proc(n) option remember; binomial(2*n, n)/ (n+1) end:

a:= n-> Determinant(Matrix(4, (i, j)-> ctln(i+j-2+n))):

seq(a(n), n=0..20);  # Alois P. Heinz, Sep 10 2008, revised, Sep 05 2019

MATHEMATICA

Join[{1}, Table[Det[Table[Binomial[i+3, j-i+4], {i, n}, {j, n}]], {n, 20}]] (* Harvey P. Dale, Jul 31 2012 *)

Table[3628800 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)!), {n, 0, 20}] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Sequence in context: A141361 A203013 A266481 * A140049 A300589 A130031

Adjacent sequences:  A006147 A006148 A006149 * A006151 A006152 A006153

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Alois P. Heinz, Sep 10 2008

STATUS

approved

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Last modified September 21 11:18 EDT 2019. Contains 327253 sequences. (Running on oeis4.)