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%I M4288 #55 Dec 20 2023 15:10:18
%S 1,1,6,91,2548,111384,6852768,553361016,55804330152,6774025632340,
%T 962310111888300,156490840602392625,28622389306817092500,
%U 5804104057179375825000,1289547073500366035700000,310827567433642575691950000,80604345356574686019872460000
%N 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}.
%C 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
%D M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique. Ph.D Dissertation, Université Bordeaux I, 1983.
%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="/A006151/b006151.txt">Table of n, a(n) for n = 0..200</a>
%H S. J. Cyvin and I. Gutman, <a href="https://doi.org/10.1007/978-3-662-00892-8_11">Kekulé structures in benzenoid hydrocarbons</a>, <a href="https://doi.org/10.1007/978-3-662-00892-8">Lecture Notes in Chemistry, No. 46</a>, Springer, New York, 1988 (see p. 183).
%H M. de Sainte-Catherine and G. Viennot, <a href="https://doi.org/10.1007/BFb0072509">Enumeration of certain Young tableaux with bounded height</a>, in: G. Labelle and P. Leroux (eds), <a href="https://doi.org/10.1007/BFb0072503">Combinatoire énumérative</a>, Lecture Notes in Mathematics, vol. 1234, Springer, Berlin, Heidelberg, 1986, pp. 58-67.
%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>, preprint, 2015.
%F From _Vaclav Kotesovec_, Mar 20 2014: (Start)
%F 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).
%F 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)!).
%F a(n) ~ 1380784741023744000 * 1024^n / (Pi^(5/2) * n^(55/2)). (End)
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j).
%F a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 10)/(i + j - 1) for n >= 1. (End)
%p 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
%p # second Maple program:
%p a:= proc(n) option remember; `if`(n=0, 1,
%p 32*mul((2*(n-i)+7)/(n+9-i), i=0..4)*a(n-1))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Mar 03 2014
%t 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_ *)
%t 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 *)
%Y Cf. A000108, A005700, A006149, A006150.
%Y Column k=5 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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