A374522 - OEIS

%I #28 Nov 08 2024 08:41:31

%S 0,3,54,657,7276,77515,807570,8288273,84146392,847289511,8476605270,

%T 84362730441,836022413604,8255176274787,81266247493186,

%U 797911337890785,7816430993273264,76417576884235999,745777615780501902,7266758081613043581,70706322844243486380,687103929058903836459

%N a(n) = n*(binomial(4*n, n) - 1).

%C For n>1, a(n) is the number of vertices of the graph for the "Left, Center, Right" game with n players.

%H Benjamin Richeson and David Richeson, <a href="https://arxiv.org/abs/2407.05069">What's the Best Seat in the Game Left, Center, Right?</a>, arXiv:2407.05069 [math.HO], 2024. See Table 1 p. 4.

%F G.f.: 4*x*hypergeom([5/4, 3/2, 7/4], [4/3, 5/3], 4^4*x/3^3) - x/(1 - x)^2. - _Stefano Spezia_, Jul 10 2024

%F D-finite with recurrence -3*(n-1)*(3*n-1)*(3*n-2)*a(n) +(283*n^3-519*n^2+398*n-144)*a(n-1) -8*(4*n-5)*(4*n-7)*(2*n-3)*a(n-2) +4*(229*n^3-783*n^2+932*n-384)=0. - _R. J. Mathar_, Jul 10 2024

%t CoefficientList[Series[4x*HypergeometricPFQ[{5/4,3/2,7/4},{4/3,5/3},256x/27]-x/(1-x)^2,{x,0,21}],x] (* _Stefano Spezia_, Jul 10 2024 *)

%o (PARI) a(n) = n*(binomial(4*n, n) - 1);

%o (Python)

%o from math import comb

%o def A374522(n): return n*(comb((n<<2),n)-1) # _Chai Wah Wu_, Jul 10 2024

%o (Magma) [n*(Binomial(4*n, n) - 1): n in [0..25]]; // _Vincenzo Librandi_, Nov 07 2024

%Y Cf. A005810.

%K nonn,easy

%O 0,2

%A _Michel Marcus_, Jul 10 2024