OFFSET
0,2
COMMENTS
Second binomial transform of aerated Fibonacci numbers.
Hankel transform is 1,1,1,-1,0,0,0,...
LINKS
Index entries for linear recurrences with constant coefficients, signature (8,-23,28,-11).
FORMULA
G.f.: (1-2x)^3/(1-8x+23x^2-28x^3+11x^4);
a(n) = Sum_{k=0..floor(n/2)} C(n,2k)*2^(n-2k)*F(k+1).
a(n) = Sum_{k=0..n} C(n,k)*2^(n-k)*F(k/2+1)*(1+(-1)^k)/2.
MATHEMATICA
LinearRecurrence[{8, -23, 28, -11}, {1, 2, 5, 14}, 30] (* Harvey P. Dale, Oct 05 2023 *)
PROG
(PARI) T(n, k) = sum(j=0, n, binomial(n, j)*binomial(n-j, 2*(k-j))*fibonacci(k-j+1));
a(n) = vecsum(vector(n+1, k, T(n, k-1))); \\ Michel Marcus, Nov 11 2022
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Jul 12 2009
STATUS
approved