A006149 Number of 3-tuples (p_1, p_2, p_3) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}. (Formerly M3634)

1, 1, 4, 30, 330, 4719, 81796, 1643356, 37119160, 922268360, 24801924512, 713055329720, 21706243125300, 694280570551875, 23188541161342500, 804601696647424500, 28880966163870711000, 1068595748063216307000, 40631980618055892780000, 1583603339463794983230000

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

Cf. A000108, A005700, A006150, A006151.

Column k=3 of A078920.

Diagonal of A123352 and of A185249.

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

Extensions

Name clarified by Alois P. Heinz, Feb 24 2023

Status

approved