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A006631 From generalized Catalan numbers.

%I A006631 M4539 #33 Sep 01 2025 03:07:02
%S A006631 1,8,52,320,1938,11704,70840,430560,2629575,16138848,99522896,
%T A006631 616480384,3834669566,23944995480,150055305008,943448717120,
%U A006631 5949850262895,37628321318280,238591135349700,1516500543586560,9660632784642840,61670325204822048,394451619337629792
%N A006631 From generalized Catalan numbers.
%D A006631 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 A006631 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006631 Vincenzo Librandi, <a href="/A006631/b006631.txt">Table of n, a(n) for n = 0..200</a>
%H A006631 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 A006631 G.f.: hypergeometric3_F_2([ 3, 8/3, 10/3 ], [ 5, 9/2 ], 27*x/4).
%F A006631 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 A006631 a(n) ~ 3^(3*n+15/2)/(2^(2n+6)*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2012
%F A006631 a(n) = 8*binomial(3*n + 8, n)/(3*n + 8). - _Andrew Howroyd_, Nov 06 2017
%t A006631 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 *)
%t A006631 Table[4*Binomial[3*n+7,n]/(n+4), {n,0,40}] (* _G. C. Greubel_, Aug 31 2025 *)
%o A006631 (PARI) a(n) = 8*binomial(3*n + 8, n)/(3*n + 8);
%o A006631 (Magma)
%o A006631 A006631:= func< n | 4*Binomial(3*n+7,n)/(n+4) >;
%o A006631 [A006631(n): n in [0..40]]; // _G. C. Greubel_, Aug 31 2025
%o A006631 (SageMath)
%o A006631 def A006631(n): return 4*binomial(3*n+7,n)//(n+4)
%o A006631 print([A006631(n) for n in range(41)]) # _G. C. Greubel_, Aug 31 2025
%Y A006631 Column 4 of A092276.
%Y A006631 Cf. A006630.
%K A006631 nonn,easy,changed
%O A006631 0,2
%A A006631 _Simon Plouffe_
%E A006631 More terms from _Vincenzo Librandi_, May 03 2013