A374522 a(n) = n*(binomial(4*n, n) - 1).

0, 3, 54, 657, 7276, 77515, 807570, 8288273, 84146392, 847289511, 8476605270, 84362730441, 836022413604, 8255176274787, 81266247493186, 797911337890785, 7816430993273264, 76417576884235999, 745777615780501902, 7266758081613043581, 70706322844243486380, 687103929058903836459

Offset: 0

Author

Michel Marcus, Jul 10 2024

Keywords

• nonn
• easy

Comments

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

Links

• Benjamin Richeson and David Richeson, What's the Best Seat in the Game Left, Center, Right?, arXiv:2407.05069 [math.HO], 2024. See Table 1 p. 4.

Crossrefs

Cf. A005810.

Programs

Magma

[n*(Binomial(4*n, n) - 1): n in [0..25]]; // Vincenzo Librandi, Nov 07 2024

Mathematica

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 *)

PARI

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

Python

from math import comb
def A374522(n): return n*(comb((n<<2),n)-1) # Chai Wah Wu, Jul 10 2024

Formula

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

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