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%I M4013 #67 Dec 10 2023 15:48:50
%S 1,1,5,55,1001,26026,884884,37119160,1844536720,105408179176,
%T 6774025632340,481155055944150,37259723952950625,3111129272480118750,
%U 277587585343361452500,26268551497229678505000,2620002484114994890890000,273961129317241857069150000,29896847445736985488399170000
%N 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}.
%C 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
%D S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 183).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A006150/b006150.txt">Table of n, a(n) for n = 0..431</a>
%H M. de Sainte-Catherine, <a href="/A006149/a006149.pdf">Couplages et Pfaffiens en Combinatoire. Physique et Informatique</a>, Ph.D Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy)
%H Nicholas M. Katz, <a href="https://web.math.princeton.edu/~nmk/catalan11.pdf">A Note on Random Matrix Integrals, Moment Identities, and Catalan Numbers</a>, 2015.
%F a(n) = Det[Table[binomial[i+3, j-i+4], {i, 1, n}, {j, 1, n}]]. - _David Callan_, Jul 20 2005
%F From _Vaclav Kotesovec_, Mar 20 2014: (Start)
%F 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).
%F 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)!).
%F a(n) ~ 14863564800 * 256^n / (Pi^2 * n^18). (End)
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j).
%F a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 8)/(i + j - 1) for n >= 1. (End)
%F E.g.f.: hypergeom([1/2, 3/2, 5/2, 7/2], [5, 6, 7, 8], 256*x). - _Stefano Spezia_, Dec 09 2023
%p with(LinearAlgebra):
%p ctln:= proc(n) option remember; binomial(2*n, n)/ (n+1) end:
%p a:= n-> Determinant(Matrix(4, (i, j)-> ctln(i+j-2+n))):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 10 2008, revised, Sep 05 2019
%t Join[{1},Table[Det[Table[Binomial[i+3,j-i+4],{i,n},{j,n}]],{n,20}]] (* _Harvey P. Dale_, Jul 31 2012 *)
%t 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 *)
%Y Cf. A000108, A005700, A006149, A006151.
%Y Column k=4 of A078920.
%Y Diagonal of A123352 and of A185249.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Alois P. Heinz_, Sep 10 2008
%E Name clarified by _Alois P. Heinz_, Feb 24 2023
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