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}. (Formerly M4288)

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

Offset

0,3

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.

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)

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 *)

Crossrefs

Cf. A000108, A005700, A006149, A006150.

Column k=5 of A078920.

Diagonal of A123352 and of A185249.

Sequence in context: A246155 A349716 A219220 * A005327 A182263 A360826

Adjacent sequences: A006148 A006149 A006150 * A006152 A006153 A006154

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