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}. (Formerly M4013)

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

Offset

0,3

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.

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

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 *)

Crossrefs

Cf. A000108, A005700, A006149, A006151.

Column k=4 of A078920.

Diagonal of A123352 and of A185249.

Sequence in context: A203013 A266481 A371316 * A373500 A140049 A300589

Adjacent sequences: A006147 A006148 A006149 * A006151 A006152 A006153

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Alois P. Heinz, Sep 10 2008

Name clarified by Alois P. Heinz, Feb 24 2023

Status

approved