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A006149 Number of Dyck paths.
(Formerly M3634)
6
1, 1, 4, 30, 330, 4719, 81796, 1643356, 37119160, 922268360, 24801924512, 713055329720, 21706243125300, 694280570551875, 23188541161342500, 804601696647424500, 28880966163870711000, 1068595748063216307000 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..17.

M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique.,  Ph.D Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy)

Nicholas M. Katz, A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers, 2015.

FORMULA

G.f.: 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

CROSSREFS

Sequence in context: A128329 A211828 A277759 * A207833 A121413 A001761

Adjacent sequences:  A006146 A006147 A006148 * A006150 A006151 A006152

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

STATUS

approved

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Last modified May 25 11:07 EDT 2018. Contains 304560 sequences. (Running on oeis4.)