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A098521 E.g.f. exp(x)*BesselI(2,2*sqrt(2)*x)/2. 2
0, 0, 1, 3, 14, 50, 195, 721, 2716, 10116, 37845, 141295, 528330, 1975766, 7395479, 27698685, 103821240, 389410568, 1461605481, 5489516955, 20630539910, 77579118330, 291893775019, 1098848179561, 4138773239892, 15596070165900, 58797332264125, 221762856917511, 836756771788098 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Binomial transform of e.g.f. BesselI(2,2*sqrt(2)*x)/2, or {0,0,1,0,8,0,60,0,448,0,3360,...} with g.f. ((1-4*x^2)-sqrt(1-8*x^2))/(8*x^2*sqrt(1-8*x^2)).

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

FORMULA

G.f.: (1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^2*sqrt(1-2*x-7*x^2)).

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, k+2)*2^k.

Conjecture: (n+2)*a(n) -(4n+3)*a(n-1) -3*(2n+1)*a(n-2) +(20n-29)*a(n-3) +21*(n-3)*a(n-4)=0. - R. J. Mathar, Dec 08 2011

Shorter recurrence (for n>=3): (n-2)*(n+2)*a(n) = n*(2*n-1)*a(n-1) + 7*(n-1)*n*a(n-2). - Vaclav Kotesovec, Dec 28 2012

a(n) ~ sqrt(8+2*sqrt(2))*(1+2*sqrt(2))^n/(8*sqrt(Pi*n)). - Vaclav Kotesovec, Dec 28 2012

MATHEMATICA

CoefficientList[Series[(1-2*x-3*x^2-(1-x)*Sqrt[1-2*x-7*x^2]) / (8*x^2*Sqrt[1-2*x-7*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Dec 28 2012 *)

PROG

(PARI) x='x+O('x^66); concat([0, 0], Vec((1-2*x-3*x^2-(1-x)*sqrt(1-2*x-7*x^2))/(8*x^2*sqrt(1-2*x-7*x^2)))) \\ Joerg Arndt, May 11 2013

(MAGMA) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); [0, 0] cat Coefficients(R!((1-2*x-3*x^2-(1-x)*Sqrt(1-2*x-7*x^2))/(8*x^2*Sqrt(1-2*x-7*x^2)))); // G. C. Greubel, Aug 17 2018

CROSSREFS

Cf. A014531, A098518.

Sequence in context: A063025 A187917 A164304 * A084150 A203196 A320826

Adjacent sequences:  A098518 A098519 A098520 * A098522 A098523 A098524

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Sep 12 2004

STATUS

approved

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Last modified June 20 20:14 EDT 2021. Contains 345232 sequences. (Running on oeis4.)