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%I M3634 #68 Feb 25 2023 08:20:28
%S 1,1,4,30,330,4719,81796,1643356,37119160,922268360,24801924512,
%T 713055329720,21706243125300,694280570551875,23188541161342500,
%U 804601696647424500,28880966163870711000,1068595748063216307000,40631980618055892780000,1583603339463794983230000
%N Number of 3-tuples (p_1, p_2, p_3) 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 3 X 3 Hankel matrix [a_0, a_1, a_2 ; a_1, a_2, a_3 ; a_2, a_3, a_4] with a_j=A000108(n+j). - _Philippe Deléham_, Apr 12 2007
%C Third subdiagonal in A123352, equivalent to the 6th subdiagonal in A185249, its "aerated" version with additional subdiagonals entirely filled with zeros. - _R. J. Mathar_, Feb 18 2011
%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 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 G. C. Greubel, <a href="/A006149/b006149.txt">Table of n, a(n) for n = 0..564</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 G.f.: Hypergeometric 4_F_3 ( [ 1, 1/2, 5/2, 3/2 ]; [ 4, 5, 6 ]; 64 x ).
%F a(n) = Det[Table[binomial[i+2, j-i+3], {i, 1, n}, {j, 1, n}]]. - _David Callan_, Jul 20 2005
%F a(n) = 720 (2*n)! (2*n+2)! (2*n+4)! / (n! (n+1)! (n+2)! (n+3)! (n+4)! (n+5)!). - _Steven Finch_, Mar 30 2008
%F (n+5)*(n+4)*(n+3)*a(n) -8*(2*n+3)*(2*n+1)*(2*n-1)*a(n-1)=0. - _R. J. Mathar_, Feb 27 2018
%F From _Peter Bala_, Feb 22 2023: (Start)
%F a(n) = Product_{1 <= i <= j <= n-1} (i + j + 6)/(i + j).
%F a(n) = (1/2^(n-1)) * Product_{1 <= i <= j <= n-1} (i + j + 6)/(i + j - 1) for n >= 1. (End)
%F a(n) ~ 45 * 2^(6*n + 10) / (Pi^(3/2) * n^(21/2)). - _Vaclav Kotesovec_, Feb 23 2023
%p seq(6!*(2*n)!*(2*n+2)!*(2*n+4)!/mul((n+j)!, j=0..5), n=0..20); # _G. C. Greubel_, Aug 28 2019
%t Table[6!*(2*n)!*(2*n+2)!*(2*n+4)!/Product[(n+j)!, {j,0,5}], {n,0,20}] (* _G. C. Greubel_, Aug 28 2019 *)
%o (PARI) vector(20, n, 6!*(2*n-2)!*(2*n)!*(2*n+2)!/prod(j=0,5, (n+j-1)!) ) \\ _G. C. Greubel_, Aug 28 2019
%o (Magma) F:=Factorial; [F(6)*F(2*n)*F(2*n+2)*F(2*n+4)/&*[F(n+j): j in [0..5]] : n in [0..20]]; // _G. C. Greubel_, Aug 28 2019
%o (Sage) f=factorial; [f(6)*f(2*n)*f(2*n+2)*f(2*n+4)/product(f(n+j) for j in (0..5)) for n in (0..20)] # _G. C. Greubel_, Aug 28 2019
%o (GAP) F:=Factorial;; List([0..20], n-> F(6)*F(2*n)*F(2*n+2)*F(2*n+4) /Product([0..5], j-> F(n+j) ) ); # _G. C. Greubel_, Aug 28 2019
%Y Cf. A000108, A005700, A006150, A006151.
%Y Column k=3 of A078920.
%Y Diagonal of A123352 and of A185249.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Name clarified by _Alois P. Heinz_, Feb 24 2023
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