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)
From Peter Bala, Feb 22 2023: (Start)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j - 1) for n >= 1. (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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Sep 10 2008
Name clarified by Alois P. Heinz, Feb 24 2023
STATUS
approved