A006150 Number of 4-tuples (p_1, p_2, ..., p_4) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

1, 1, 5, 55, 1001, 26026, 884884, 37119160, 1844536720, 105408179176, 6774025632340, 481155055944150, 37259723952950625, 3111129272480118750, 277587585343361452500, 26268551497229678505000, 2620002484114994890890000, 273961129317241857069150000, 29896847445736985488399170000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000108, A005700, A006149, A006151.

Column k=4 of A078920.

Diagonal of A123352 and of A185249.

Programs

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 *)

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

Extensions

More terms from Alois P. Heinz, Sep 10 2008

Name clarified by Alois P. Heinz, Feb 24 2023