A219314 Composition of the inverse binomial transform of Fibonacci numbers and the Catalan transform of Fibonacci numbers.
0, 1, 0, 3, 3, 13, 26, 77, 192, 529, 1412, 3873, 10603, 29315, 81318, 226763, 634627, 1782637, 5022840, 14193457, 40211105, 114191159, 324981030, 926720807, 2647513282, 7576475383, 21716189676, 62336237007, 179182653117, 515717424109, 1486119467026
Offset: 0
Links
- Fung Lam, Table of n, a(n) for n = 0..2000
- Paul Barry, A Catalan transform and related transformations on integer sequences, pp. 20-22.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).
Formula
G.f.: ((1+2*x)*sqrt(1-2*x-3*x^2) - 1 + x + 2*x^2)/(2*(1+x)*(1-2*x-4*x^2)).
Asymptotics: a(n) ~ 3^(n+2)*5/(8*sqrt(3*Pi*n^3)). - Fung Lam, Apr 07 2014
Conjecture: n*a(n) -2*n*a(n-1) +11*(-n+2)*a(n-2) +4*(2*n-5)*a(n-3) +8*(5*n-17)*a(n-4) +24*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 14 2016
Conjecture: n*(5*n-7)*a(n) -4*(5*n^2-12*n+6)*a(n-1) -(15*n^2-11*n-30) *a(n-2) +2*(35*n^2-119*n+66)*a(n-3) +12*(n-3)*(5*n2)*a(n-4)=0. - R. J. Mathar, Jun 14 2016