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).
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
Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique, PhD Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy of pages III.42-III.45)
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)
From Peter Bala, Feb 22 2023: (Start)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j - 1) for n >= 1. (End)
E.g.f.: hypergeom([1/2, 3/2, 5/2, 7/2], [5, 6, 7, 8], 256*x). - Stefano Spezia, Dec 09 2023
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
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