OFFSET
0,3
COMMENTS
a(n) is the determinant of the 3 X 3 Hankel matrix [a_0, a_1, a_2 ; a_1, a_2, a_3 ; a_2, a_3, a_4] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007
Third subdiagonal in A123352, equivalent to the 6th subdiagonal in A185249, its "aerated" version with additional subdiagonals entirely filled with zeros. - R. J. Mathar, Feb 18 2011
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..564
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
G.f.: Hypergeometric 4_F_3 ( [ 1, 1/2, 5/2, 3/2 ]; [ 4, 5, 6 ]; 64 x ).
a(n) = Det[Table[binomial[i+2, j-i+3], {i, 1, n}, {j, 1, n}]]. - David Callan, Jul 20 2005
a(n) = 720 (2*n)! (2*n+2)! (2*n+4)! / (n! (n+1)! (n+2)! (n+3)! (n+4)! (n+5)!). - Steven Finch, Mar 30 2008
(n+5)*(n+4)*(n+3)*a(n) -8*(2*n+3)*(2*n+1)*(2*n-1)*a(n-1)=0. - R. J. Mathar, Feb 27 2018
From Peter Bala, Feb 22 2023: (Start)
a(n) = Product_{1 <= i <= j <= n-1} (i + j + 6)/(i + j).
a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 6)/(i + j - 1) for n >= 1. (End)
a(n) ~ 45 * 2^(6*n + 10) / (Pi^(3/2) * n^(21/2)). - Vaclav Kotesovec, Feb 23 2023
MAPLE
seq(6!*(2*n)!*(2*n+2)!*(2*n+4)!/mul((n+j)!, j=0..5), n=0..20); # G. C. Greubel, Aug 28 2019
MATHEMATICA
Table[6!*(2*n)!*(2*n+2)!*(2*n+4)!/Product[(n+j)!, {j, 0, 5}], {n, 0, 20}] (* G. C. Greubel, Aug 28 2019 *)
PROG
(PARI) vector(20, n, 6!*(2*n-2)!*(2*n)!*(2*n+2)!/prod(j=0, 5, (n+j-1)!) ) \\ G. C. Greubel, Aug 28 2019
(Magma) F:=Factorial; [F(6)*F(2*n)*F(2*n+2)*F(2*n+4)/&*[F(n+j): j in [0..5]] : n in [0..20]]; // G. C. Greubel, Aug 28 2019
(Sage) f=factorial; [f(6)*f(2*n)*f(2*n+2)*f(2*n+4)/product(f(n+j) for j in (0..5)) for n in (0..20)] # G. C. Greubel, Aug 28 2019
(GAP) F:=Factorial;; List([0..20], n-> F(6)*F(2*n)*F(2*n+2)*F(2*n+4) /Product([0..5], j-> F(n+j) ) ); # G. C. Greubel, Aug 28 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Name clarified by Alois P. Heinz, Feb 24 2023
STATUS
approved