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%I M4539 #27 Feb 25 2018 22:51:42
%S 1,8,52,320,1938,11704,70840,430560,2629575,16138848,99522896,
%T 616480384,3834669566,23944995480,150055305008,943448717120,
%U 5949850262895,37628321318280,238591135349700,1516500543586560,9660632784642840,61670325204822048,394451619337629792
%N From generalized Catalan numbers.
%D H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vincenzo Librandi, <a href="/A006631/b006631.txt">Table of n, a(n) for n = 0..200</a>
%H Emanuele Munarini, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Munarini/muna4.html">Shifting Property for Riordan, Sheffer and Connection Constants Matrices</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
%F G.f.: 3_F_2 ( [ 3, 8/3, 10/3 ]; [ 5, 9/2 ]; 27 x / 4 ).
%F Recurrence: 2*(n+4)*(2*n+7)*a(n) = (5*n+13)*(11*n+29)*a(n-1) - 7*(31*n^2+87*n+62)*a(n-2) + 21*(3*n-1)*(3*n+1)*a(n-3). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) ~ 3^(3n+15/2)/(2^(2n+6)*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F a(n) = 8*binomial(3*n + 8, n)/(3*n + 8). - _Andrew Howroyd_, Nov 06 2017
%t Table[SeriesCoefficient[HypergeometricPFQ[{3,8/3,10/3},{5,9/2},27*x/4],{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 07 2012 *)
%o (PARI) a(n) = 8*binomial(3*n + 8, n)/(3*n + 8);
%Y Column 4 of A092276.
%K nonn,easy
%O 0,2
%A _Simon Plouffe_
%E More terms from _Vincenzo Librandi_, May 03 2013
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