A349695 a(n) = (2n^2 - n + 2) * (2n)! / ((n + 1) * (n + 2) * n!^2).

1, 4, 17, 70, 282, 1122, 4433, 17446, 68510, 268736, 1053626, 4130524, 16195220, 63517950, 249214545, 978228870, 3841579830, 15093382920, 59329549950, 233324644020, 918016888620, 3613561263780, 14230112774202, 56061218710620, 220948453132652, 871137317136352

Offset

1,2

Comments

This sequence arises in the study of subleading corrections (in the large N-expansion) related to quantities in the Superconformal Field Theory called "N=4 SYM" with gauge group SU(N).

Links

Table of n, a(n) for n=1..26.

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998.

Formula

G.f.: (8*x^2-9*x+2)/(2*x^2*sqrt(1-4*x)) - 1.

a(n) = A000108(n)*(2n^2-n+2)/(n+2).

E.g.f.: exp(2*x)*(3*BesselI(0, 2*x) - (2 + x)*F(2, x^2)) - 1, where F is the regularized confluent hypergeometric function. - Stefano Spezia, Nov 27 2021

a(n) = 4^n*(2*n^2-n+2)*Gamma(n+1/2)/(sqrt(Pi)*(n+2)!). - Chai Wah Wu, Dec 17 2021

D-finite with recurrence (n+2)*(2*n^2-5*n+5)*a(n) -2*(2*n-1)*(2*n^2-n+2)*a(n-1)=0. - R. J. Mathar, Mar 06 2022

Mathematica

nterms=40; Table[(2n^2-n+2)(2n)!/((n+1)(n+2)n!^2), {n, nterms}] (* Paolo Xausa, Nov 25 2021 *)
CoefficientList[Series[Exp[2x](3BesselI[0, 2x]-(2+x)Hypergeometric0F1Regularized[2, x^2])-1, {x, 0, 26}], x]Table[n!, {n, 0, 26}] (* Stefano Spezia, Nov 27 2021 *)

Prog

(PARI) a(n) = (2*n^2 - n + 2)*(2*n)! / ((n + 1)*(n + 2)*n!^2); \\ Michel Marcus, Nov 27 2021

Crossrefs

Cf. A000108.

Sequence in context: A330246 A339065 A005511 * A003127 A172447 A291009

Adjacent sequences: A349692 A349693 A349694 * A349696 A349697 A349698

Keyword

nonn,easy

Author

Alessandro Pini, Nov 25 2021

Status

approved