|
| |
|
|
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
|
| |
|
|
