A006151 Number of 5-tuples (p_1, p_2, ..., p_5) of Dyck paths of semilength n, such that each p_i is never below p_{i-1}.

1, 1, 6, 91, 2548, 111384, 6852768, 553361016, 55804330152, 6774025632340, 962310111888300, 156490840602392625, 28622389306817092500, 5804104057179375825000, 1289547073500366035700000, 310827567433642575691950000, 80604345356574686019872460000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

a(n) is the determinant of the 5 X 5 Hankel matrix [a_0, a_1, a_2, a_3, a_4 ; a_1, a_2, a_3, a_4, a_5 ; a_2, a_3, a_4, a_5, a_6 ; a_3, a_4, a_5, a_6, a_7 ; a_4, a_5, a_6, a_7, a_8] with a_j=A000108(n+j). - Philippe Deléham, Apr 12 2007

References

• 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

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• 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 and G. Viennot, Enumeration of certain Young tableaux with bounded height, in: G. Labelle and P. Leroux (eds), Combinatoire énumérative, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
• Nicholas M. Katz, A note on random matrix integrals, moment identities, and Catalan numbers, preprint, 2015.

Crossrefs

Cf. A000108, A005700, A006149, A006150.

Column k=5 of A078920.

Diagonal of A123352 and of A185249.

Programs

Maple

with(linalg): ctln:= proc(n) option remember; binomial(2*n,n)/(n+1) end: a:= n-> det(Matrix(5, (i,j)-> ctln(i+j-2+n))): seq(a(n), n=0..20); # Alois P. Heinz, Sep 10 2008
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
       32*mul((2*(n-i)+7)/(n+9-i), i=0..4)*a(n-1))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 03 2014

Mathematica

a[n_] := Det[Array[CatalanNumber[#1 + #2 - 2 + n]&, {5, 5}]]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
Table[1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!),{n,0,20}] (* Vaclav Kotesovec, Mar 20 2014 *)

Formula

From Vaclav Kotesovec, Mar 20 2014: (Start)

Recurrence: (n+5)*(n+6)*(n+7)*(n+8)*(n+9)*a(n) = 32*(2*n-1)*(2*n+1)*(2*n+3)*(2*n+5)*(2*n+7)*a(n-1).

a(n) = 1316818944000 * (2*n)! * (2*(n+1))! * (2*(n+2))! * (2*(n+3))! * (2*(n+4))! / (n! * (n+1)! * (n+2)! * (n+3)! * (n+4)! * (n+5)! * (n+6)! * (n+7)! * (n+8)! * (n+9)!).

a(n) ~ 1380784741023744000 * 1024^n / (Pi^(5/2) * n^(55/2)). (End)

From Peter Bala, Feb 22 2023: (Start)

a(n) = Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j).

a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j - 1) for n >= 1. (End)

Extensions

More terms from Alois P. Heinz, Sep 10 2008

Name clarified by Alois P. Heinz, Feb 24 2023