A124126 a(n)=(1/(3n))*sum(k=1,n,F(8k)*B(2n-2k)*binomial(2n,2k)) where F=Fibonacci numbers and B=Bernoulli numbers.

7, 168, 5425, 199367, 7890120, 327681361, 14071534535, 618924449640, 27702229113265, 1255905441590279, 57477374413516680, 2648841480448502353, 122698149590393354375, 5704992303566275023912, 265994788806640480586545

Offset

1,1

Comments

Linear recurrence and empirical g.f. confirmed with more terms. - Ray Chandler, Mar 07 2024

Links

Ray Chandler, Table of n, a(n) for n = 1..599

Index entries for linear recurrences with constant coefficients, signature (132, -6594, 157376, -1866219, 10372992, -22605346, 11987460, -245025).

Formula

a(n)=(1/(3n))*(F(8n-4)+2*L(4n-2)*5^(n-1)+2*F(2n-1)*3^(2n-1)+U(n)) where L=Lucas numbers and U(n) satisfies the order 2 recursion : U(1)=2, U(2)=24, U(n)=23U(n-1)-121U(n-2).

Empirical g.f.: x*(48015*x^7 +9278012*x^6 -12039433*x^5 +3970491*x^4 -510573*x^3 +29407*x^2 -756*x +7) / ((x^2 -47*x +1)*(25*x^2 -35*x +1)*(81*x^2 -27*x +1)*(121*x^2 -23*x +1)). - Colin Barker, Jun 28 2013

Prog

(PARI) a(n)=(1/3/n)*sum(k=1, n, fibonacci(8*k)*bernfrac(2*n-2*k)*binomial(2*n, 2*k))

Crossrefs

Cf. A111262.

Sequence in context: A009807 A220991 A165388 * A086373 A162131 A258299

Adjacent sequences: A124123 A124124 A124125 * A124127 A124128 A124129

Keyword

nonn

Author

Benoit Cloitre, Nov 29 2006

Status

approved