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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.

Nicholas M. Katz, A NOTE ON RANDOM MATRIX INTEGRALS, MOMENT IDENTITIES, AND CATALAN NUMBERS, 2015; https://web.math.princeton.edu/~nmk/catalan11.pdf

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..140

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(linalg): ctln:= proc(n) option remember; binomial(2*n, n)/ (n+1) end: a:= n-> det(Matrix(4, (i, j)-> ctln(i+j-2+n))): seq(a(n), n=0..20);  # Alois P. Heinz, Sep 10 2008

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 A130031 A119399

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 August 21 11:52 EDT 2017. Contains 290864 sequences.