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}.

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

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000108, A005700, A006150, A006151.

Column k=3 of A078920.

Diagonal of A123352 and of A185249.

Programs

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

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

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 *)

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

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

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

Extensions

Name clarified by Alois P. Heinz, Feb 24 2023