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A278472 a(n) = Sum_{i=0..n} Fibonacci(i+1)*binomial(2*n-i+2, n+2). 1
1, 5, 22, 92, 376, 1518, 6085, 24285, 96647, 383911, 1523117, 6037745, 23920853, 94737897, 375125126, 1485173396, 5879740780, 23277813786, 92160762514, 364906983652, 1444972555742, 5722488162840, 22665368420672, 89783494878902 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

FORMULA

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

a(n) ~ 2^(2*n+4)/sqrt(Pi*n). - Vaclav Kotesovec, Nov 23 2016

Conjecture: +(n+2)*(n^2-4*n+1)*a(n) +2*(-4*n^3+9*n^2+16*n-6)*a(n-1) +(15*n^3-38*n^2-9*n+14)*a(n-2) +2*(2*n-1)*(n^2-2*n-2)*a(n-3)=0. - R. J. Mathar, Mar 06 2017

Conjecture: +(n+2)*a(n) +(-11*n-14)*a(n-1) +(37*n+20)*a(n-2) +(-25*n-6)*a(n-3) +2*(-21*n+41)*a(n-4) +4*(-2*n+5)*a(n-5)=0. - R. J. Mathar, Mar 06 2017

MATHEMATICA

Table[Sum[Fibonacci[j+1]*Binomial[2*n-j+2, n+2], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 23 2016 *)

FullSimplify[Table[Binomial[2*n+2, n+2]*(Hypergeometric2F1[1, -n, -2*(n+1), -1/GoldenRatio] + GoldenRatio^2 * Hypergeometric2F1[1, -n, -2*(n+1), GoldenRatio])/(Sqrt[5]*GoldenRatio), {n, 0, 20}]] (* Vaclav Kotesovec, Nov 23 2016 *)

PROG

(Maxima)

taylor(-(2*x+sqrt(1-4*x)-1)/(2*sqrt(1-4*x)*x-8*x+2)/x^2, x, 0, 20)

(PARI) x='x+O('x^50); Vec(-(2*x+sqrt(1-4*x)-1)/((2*sqrt(1-4*x)*x-8*x+2)*x^2)) \\ G. C. Greubel, Jun 07 2017

(PARI) a(n) = sum(i=0, n, fibonacci(i+1)*binomial(2*n-i+2, n+2)); \\ Michel Marcus, Jun 06 2017

CROSSREFS

Cf. A000045.

Sequence in context: A211973 A053297 A071715 * A010036 A127617 A095932

Adjacent sequences:  A278469 A278470 A278471 * A278473 A278474 A278475

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, Nov 23 2016

STATUS

approved

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Last modified October 21 09:24 EDT 2018. Contains 316407 sequences. (Running on oeis4.)