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%I A278472 #23 Jun 07 2017 00:42:53
%S A278472 1,5,22,92,376,1518,6085,24285,96647,383911,1523117,6037745,23920853,
%T A278472 94737897,375125126,1485173396,5879740780,23277813786,92160762514,
%U A278472 364906983652,1444972555742,5722488162840,22665368420672,89783494878902
%N A278472 a(n) = Sum_{i=0..n} Fibonacci(i+1)*binomial(2*n-i+2, n+2).
%H A278472 G. C. Greubel, <a href="/A278472/b278472.txt">Table of n, a(n) for n = 0..1000</a>
%F A278472 G.f.: -(2*x+sqrt(1-4*x)-1)/((2*sqrt(1-4*x)*x-8*x+2)*x^2).
%F A278472 a(n) ~ 2^(2*n+4)/sqrt(Pi*n). - _Vaclav Kotesovec_, Nov 23 2016
%F A278472 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
%F A278472 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
%t A278472 Table[Sum[Fibonacci[j+1]*Binomial[2*n-j+2,n+2],{j,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Nov 23 2016 *)
%t A278472 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 *)
%o A278472 (Maxima)
%o A278472 taylor(-(2*x+sqrt(1-4*x)-1)/(2*sqrt(1-4*x)*x-8*x+2)/x^2,x,0,20)
%o A278472 (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
%o A278472 (PARI) a(n) = sum(i=0, n, fibonacci(i+1)*binomial(2*n-i+2, n+2)); \\ _Michel Marcus_, Jun 06 2017
%Y A278472 Cf. A000045.
%K A278472 nonn
%O A278472 0,2
%A A278472 _Vladimir Kruchinin_, Nov 23 2016

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