A098659 Expansion of 1/sqrt((1-7*x)^2-24*x^2).

1, 7, 61, 595, 6145, 65527, 712909, 7863667, 87615745, 983726695, 11112210781, 126142119187, 1437751935361, 16443380994775, 188609259215725, 2168833084841395, 24994269200292865, 288596644195946695, 3337978523215692925, 38666734085509918675

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.

Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

Rob Noble, Asymptotics of a family of binomial sums, J. Number Theory 130 (2010), no. 11, 2561-2585.

Formula

G.f.: 1/sqrt(1-14*x+25*x^2).

E.g.f.: exp(7*x)*BesselI(0, 2*sqrt(6)*x).

a(n) = Sum_{k=0..n} C(n, k)^2*6^k.

a(n) = [x^n] (1+7*x+6*x^2)^n.

From Vaclav Kotesovec, Sep 15 2012: (Start)

General recurrence for Sum_{k=0..n} C(n,k)^2*x^k (this is case x=6): (n+2)*a(n+2)-(x+1)*(2*n+3)*a(n+1)+(x-1)^2*(n+1)*a(n)=0.

Asymptotic (Rob Noble, 2010): a(n) ~ (1+sqrt(x))^(2*n+1)/(2*x^(1/4)*sqrt(Pi*n)), this is case x=6.

(End)

D-finite with recurrence: n*a(n) +7*(-2*n+1)*a(n-1) +25*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 20 2020

Mathematica

Table[Sum[Binomial[n, k]^2*6^k, {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Sep 15 2012 *)
CoefficientList[Series[1/Sqrt[(1-7*x)^2-24*x^2], {x, 0, 25}], x] (* Stefano Spezia, Dec 04 2018 *)

Prog

(PARI) x='x+O('x^66); Vec(1/sqrt(1-14*x+25*x^2)) \\ Joerg Arndt, May 12 2013

Crossrefs

Cf. A000984, A001850, A069835, A084771, A084772.

Sequence in context: A066443 A108448 A218473 * A269731 A199686 A113718

Adjacent sequences: A098656 A098657 A098658 * A098660 A098661 A098662

Keyword

easy,nonn

Author

Paul Barry, Sep 20 2004

Status

approved