OFFSET
0,3
COMMENTS
Transform of the Fibonacci numbers under the Riordan array (1/(1-x)^2,x(1-x)^2)) (convolution array of natural numbers).
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5, -7, 5, -1).
FORMULA
a(n) = 5*a(n-1) - 7*a(n-2) + 5*a(n-3) - a(n-4).
a(n) = Sum_{k=0..n} binomial(n+k+1, 2*k+1)*F(k), where F(n) = Fibonacci(n).
a(n) = -a(-2-n) for all n in Z.
EXAMPLE
G.f. = x + 5*x^2 + 18*x^3 + 60*x^4 + 198*x^5 + 655*x^6 + 2171*x^7 + 7200*x^8 + ... - Michael Somos, Aug 12 2018
MATHEMATICA
CoefficientList[Series[x/(1-5x+7x^2-5x^3+x^4), {x, 0, 30}], x] (* or *) LinearRecurrence[{5, -7, 5, -1}, {0, 1, 5, 18}, 30] (* Harvey P. Dale, Sep 14 2013 *)
Table[Sum[Binomial[n+k+1, 2*k+1]*Fibonacci[k], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Aug 12 2018 *)
a[ n_] := Sign[n + 1] SeriesCoefficient[ x / (1 - 5 x + 7 x^2 - 5 x^3 + x^4), {x, 0, Max[n, -2 - n]}]; (* Michael Somos, Aug 12 2018 *)
PROG
(PARI) x='x+O('x^50); concat([0], Vec(x/(1-5*x+7*x^2-5*x^3+x^4))) \\ G. C. Greubel, Aug 12 2018
(PARI) for(n=0, 50, print1(sum(k=0, n, binomial(n+k+1, 2*k+1)*fibonacci(k)), ", ")) \\ G. C. Greubel, Aug 12 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x/(1-5*x+7*x^2-5*x^3+x^4))); // G. C. Greubel, Aug 12 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 17 2005
STATUS
approved