A279014 a(n) = Sum_{k=0..n} fibonacci(k+1)*binomial(2*n-1,n-k).

1, 2, 8, 33, 138, 581, 2455, 10395, 44068, 186953, 793453, 3368279, 14300161, 60713627, 257763847, 1094294875, 4645306802, 19717723173, 83687094899, 355155267179, 1507078468075, 6394577650959, 27129846069301, 115091608301743

Offset

0,2

Links

Indranil Ghosh, Table of n, a(n) for n = 0..100

Formula

G.f.: (2*x)/(((-(1-sqrt(1-4*x))/(1+sqrt(1-4*x))-((1-sqrt(1-4*x))/(1+sqrt(1-4*x)))^2)+1)*(1-sqrt(1-4*x))*sqrt(1-4*x)).

Conjecture: +n*(3*n-62)*a(n) +(3*n^2+362*n-247)*a(n-1) +(-171*n^2+220*n+162)*a(n-2) +(417*n^2-2570*n+3551)*a(n-3) +2*(27*n-59)*(2*n-7)*a(n-4)=0. - R. J. Mathar, Mar 12 2017

Conjecture: +n*(n^2-11*n+22)*a(n) +2*(-4*n^3+45*n^2-113*n+60)*a(n-1) +(15*n^3-173*n^2+530*n-480)*a(n-2) +2*(2*n-5)*(n^2-9*n+12)*a(n-3)=0. - R. J. Mathar, Mar 12 2017

a(n) ~ phi^(3*n) / sqrt(5), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Jul 10 2021

Prog

(Maxima)
taylor((2*x)/(((-(1-sqrt(1-4*x))/(1+sqrt(1-4*x))-((1-sqrt(1-4*x))/(1+sqrt(1-4*x)))^2)+1)*(1-sqrt(1-4*x))*sqrt(1-4*x)), x, 0, 27)
(Python)
from sympy import binomial, fibonacci
def a(n): return sum([fibonacci(k + 1)*binomial(2*n - 1, n - k) for k in range(n + 1)])
print([a(n) for n in range(24)]) # Indranil Ghosh, Jun 30 2017
(PARI) a(n) = sum(k=0, n, fibonacci(k+1)*binomial(2*n-1, n-k)); \\ Michel Marcus, Jun 30 2017

Crossrefs

Cf. A000045.

Sequence in context: A037513 A037716 A373752 * A099015 A150865 A150866

Adjacent sequences: A279011 A279012 A279013 * A279015 A279016 A279017

Keyword

nonn

Author

Vladimir Kruchinin, Dec 03 2016

Status

approved