cp's OEIS Frontend

A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).

%I A006634 M4648 #50 Sep 04 2025 00:48:47
%S A006634 1,9,72,570,4554,36855,302064,2504304,20974005,177232627,1509395976,
%T A006634 12943656180,111676661460,968786892675,8445123522144,73940567860896,
%U A006634 649942898236596,5733561315124260,50744886833898400,450461491952952690,4009721145437152530,35782256673785401065
%N A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).
%C A006634 Former name: From generalized Catalan numbers.
%D A006634 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 A006634 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006634 Vincenzo Librandi, <a href="/A006634/b006634.txt">Table of n, a(n) for n = 0..200</a>
%H A006634 Simon Plouffe, <a href="http://arxiv.org/abs/0911.4975">Approximations of generating functions and a few conjectures</a>, arXiv:0911.4975 [math.NT], 2002.
%F A006634 G.f.: hypergeom([9/4, 5/2, 11/4, 3], [10/3, 11/3, 4], 256/27*x). - _Simon Plouffe_, Master's Thesis, UQAM, 1992
%F A006634 G.f.: g^9 where g = 1+x*g^4 is the g.f. of A002293. - _Mark van Hoeij_, Apr 22 2013
%F A006634 From _G. C. Greubel_, Sep 01 2025: (Start)
%F A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).
%F A006634 E.g.f.: hypergeom([9/4, 5/2, 11/4, 3], [1, 10/3, 11/3, 4], 256*x/27). (End)
%p A006634 series(RootOf(g = 1+x*g^4,g)^9, x=0, 30); # _Mark van Hoeij_, Apr 22 2013
%t A006634 f[x_] := HypergeometricPFQ[ {9/4, 5/2, 11/4, 3}, {10/3, 11/3, 4}, 256/27*x]; Series[f[x], {x, 0, 16}] // CoefficientList[#, x]& (* _Jean-François Alcover_, Apr 23 2013, after _Simon Plouffe_ *)
%t A006634 Table[3*Binomial[4*n+8,n]/(n+3), {n,0,40}] (* _G. C. Greubel_, Sep 01 2025 *)
%o A006634 (PARI)
%o A006634 N = 3*66;  x = 'x + O('x^N);
%o A006634 g=serreverse(x-x^4)/x;
%o A006634 gf=g^9;  v=Vec(gf);
%o A006634 vector(#v\3,n,v[3*n-2])
%o A006634 /* _Joerg Arndt_, Apr 23 2013 */
%o A006634 (Magma)
%o A006634 A006634:= func< n | 3*Binomial(4*n+8,n)/(n+3) >;
%o A006634 [A006634(n): n in [0..40]]; // _G. C. Greubel_, Sep 01 2025
%o A006634 (SageMath)
%o A006634 def A006634(n): return 3*binomial(4*(n+2),n)//(n+3)
%o A006634 print([A006634(n) for n in range(41)]) # _G. C. Greubel_, Sep 01 2025
%Y A006634 Cf. A006630, A006631, A006632, A006633, A006635, A006636, A006637.
%K A006634 nonn,easy,changed
%O A006634 0,2
%A A006634 _Simon Plouffe_
%E A006634 More terms from _Joerg Arndt_, Apr 23 2013
%E A006634 New name by _G. C. Greubel_, Sep 01 2025