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A006151 Number of Dyck paths.
(Formerly M4288)
4
1, 1, 6, 91, 2548, 111384, 6852768, 553361016, 55804330152, 6774025632340, 962310111888300, 156490840602392625, 28622389306817092500, 5804104057179375825000, 1289547073500366035700000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

a(n) is the determinant of the 5 X 5 Hankel matrix [a_0, a_1, a_2, a_3, a_4 ; a_1, a_2, a_3, a_4, a_5 ; a_2, a_3, a_4, a_5, a_6 ; a_3, a_4, a_5, a_6, a_7 ; a_4, a_5, a_6, a_7, a_8] 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..200

FORMULA

From Vaclav Kotesovec, Mar 20 2014: (Start)

Recurrence: (n+5)*(n+6)*(n+7)*(n+8)*(n+9)*a(n) = 32*(2*n-1)*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)*a(n-1).

a(n) = 1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!).

a(n) ~ 1380784741023744000 * 1024^n / (Pi^(5/2) * n^(55/2)).

(End)

MAPLE

with(linalg): ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end: a:= n-> det(Matrix(5, (i, j)-> ctln(i+j-2+n))): seq(a(n), n=0..20); # Alois P. Heinz, Sep 10 2008

# second Maple program:

a:= proc(n) option remember; `if`(n=0, 1,

       32*mul((2*(n-i)+7)/(n+9-i), i=0..4)*a(n-1))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Mar 03 2014

MATHEMATICA

a[n_] := Det[Array[CatalanNumber[#1 + #2 - 2 + n]&, {5, 5}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)

Table[1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!), {n, 0, 20}] (* Vaclav Kotesovec, Mar 20 2014 *)

CROSSREFS

Sequence in context: A095864 A246155 A219220 * A005327 A182263 A171910

Adjacent sequences:  A006148 A006149 A006150 * A006152 A006153 A006154

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 March 23 21:31 EDT 2017. Contains 283985 sequences.