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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, 310827567433642575691950000, 80604345356574686019872460000 (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

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..200

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 and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.

Nicholas M. Katz, A note on random matrix integrals, moment identities, and Catalan numbers, preprint, 2015.

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: A246155 A349716 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 January 31 09:04 EST 2023. Contains 359970 sequences. (Running on oeis4.)