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A093131 Binomial transform of Fibonacci(2n). 8
0, 1, 5, 20, 75, 275, 1000, 3625, 13125, 47500, 171875, 621875, 2250000, 8140625, 29453125, 106562500, 385546875, 1394921875, 5046875000, 18259765625, 66064453125, 239023437500, 864794921875, 3128857421875, 11320312500000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Second binomial transform of Fibonacci(n). - Paul Barry, Apr 22 2005

LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1791

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

M. Griffiths, Families of Sequences From a Class of Multinomial Sums, Journal of Integer Sequences, 15 (2012), #12.1.8.

J. Pan, Multiple Binomial Transforms and Families of Integer Sequences , J. Int. Seq. 13 (2010), 10.4.2, F^(2) and absolute values of F^(-2).

J. Pan, Some Properties of the Multiple Binomial Transform and the Hankel Transform of Shifted Sequences , J. Int. Seq. 14 (2011) # 11.3.4, remark 14.

Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).

Index entries for linear recurrences with constant coefficients, signature (5,-5).

FORMULA

G.f.: x/(1 - 5*x + 5*x^2).

a(n) = (((5 + sqrt(5))/2)^n - ((5 - sqrt(5))/2)^n)/sqrt(5).

a(n) = A093130(n)/2^n.

a(n) = Sum_{k=0..n} Sum_{j=0..n} C(n, j)*C(j, k)*Fibonacci(j-k). - Paul Barry, Feb 15 2005

a(n) = Sum_{k=0..n} C(n, k)*2^k*Fibonacci(n-k) = Sum_{k=0..n} C(n, k)*2^(n-k) * Fibonacci(k). - Paul Barry, Apr 22 2005

a(n) = A030191(n-1), n > 0. - R. J. Mathar, Sep 05 2008

E.g.f.: 2*exp(5*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 11 2017

From Kai Wang, Dec 22 2019: (Start)

a(n) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} 5^i*((i+j)!/(i!*j!)).

a(n*k)/a(k) = Sum_{i=0..n-1; j=0..n-1; i+2*j=n-1} (-1)^(j*(k-1))*b(k)^i*((i+j)!/(i!*j!)).

a((2*m+1)*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i)*k) + 5^(m*k).

a(2*m*k)/a(k) = Sum_{i=0..m-1} (-1)^(i*k)*A020876((2*m-2*i-1)*k}.

a(m+r)*a(n+s) - a(m+s)*a(n+r) = -5^(n+s)*a(m-n)*a(r-s).

a(m+r)*a(n+s) + a(m+s)*a(n+r) = (2*A020876(m+n+r+s) - 5^(n+s)*A020876(m-n)*A020876(r-s))/5.

A020876(m+r)*A020876(n+s) - A020876(m+s)*A020876(n+r) = 5^(n+s+1)*a(m-n)*a(r-s).

A020876(m+r)*A020876(n+s) - 5*a(m+s)*a(n+r) = 5^(n+s)*A020876(m-n)*A020876(r-s).

A020876(m+r)*A020876(n+s) + 5*a(m+s)*a(n+r) = 2*A020876(m+n+r+s) + 5^(n+s+1)*a(m-n)*a(r-s).

a(n)^2 - a(n+1)*a(n-1) = 5^(n-1).

a(n)^2 - a(n+r)*a(n-r) = 5^(n-r)*a(r)^2.

a(m)*a(n+1) - a(m+1)*a(n) = 5^n*a(m-n).

a(m-n) = (a(m)*A020876(n) - A020876(m)*a(n))/(2*5^n).

a(m+n) = (a(m)*A020876(n) + A020876(m)*a(n))/2.

A020876(n)^2 - A020876(n+r)*A020876(n-r) = -5^(n-r+1)*a(r)^2.

A020876(m)*A020876(n+1) - A020876(m+1)*A020876(n) = -5^(n+1)*a(m-n).

A020876(m+n) - 5^(n)*A020876(m-n) = 5*a(m)*a(n).

A020876(m-n) = (A020876(m)*A020876(n) - 5*a(m)*a(n))/(2*5^n).

A020876(m+n) = (A020876(m)*A020876(n) + 5*a(m)*a(n))/2.  (End)

a(2*n) = 5^n*Fibonacci(2*n), a(2*n+1) = 5^n*Lucas(2*n+1). - G. C. Greubel, Dec 27 2019

MAPLE

seq(coeff(series(x/(1-5*x+5*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 27 2019

MATHEMATICA

CoefficientList[Series[x/(1-5x+5x^2), {x, 0, 30}], x] (* Michael De Vlieger, Dec 22 2019 *)

Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n], 5^((n-1)/2)*LucasL[n]], {n, 0, 30}] (* G. C. Greubel, Dec 27 2019 *)

PROG

(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-5*x+5*x^2))) \\ G. C. Greubel, Dec 27 2019

(MAGMA) I:=[0, 1]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Dec 27 2019

(Sage)

def A093131_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( x/(1-5*x+5*x^2) ).list()

A093131_list(30) # G. C. Greubel, Dec 27 2019

(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # G. C. Greubel, Dec 27 2019

CROSSREFS

Cf. A000032, A000045, A020876, A030191.

Sequence in context: A022633 A092490 A094828 * A030191 A224422 A000344

Adjacent sequences:  A093128 A093129 A093130 * A093132 A093133 A093134

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 23 2004

STATUS

approved

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Last modified March 8 01:41 EST 2021. Contains 341934 sequences. (Running on oeis4.)