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A093131 Binomial transform of Fibonacci(2n). 6
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

Table of n, a(n) for n=0..24.

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.

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

FORMULA

G.f.: x/(1 - 5x + 5x^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)F(j-k). - Paul Barry, Feb 15 2005

a(n) = Sum_{k=0..n} C(n, k)F(n-k)2^k = Sum_{k=0..n} C(n, k)F(k)2^(n-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

CROSSREFS

Cf. A000045, 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 October 16 00:50 EDT 2018. Contains 316252 sequences. (Running on oeis4.)