A090826 - OEIS

A090826

Convolution of Catalan and Fibonacci numbers.

0, 1, 2, 5, 12, 31, 85, 248, 762, 2440, 8064, 27300, 94150, 329462, 1166512, 4170414, 15031771, 54559855, 199236416, 731434971, 2697934577, 9993489968, 37157691565, 138633745173, 518851050388, 1947388942885, 7328186394725

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OFFSET

0,3

COMMENTS

Also (with a(0)=1 instead of 0): Number of fixed points in range [A014137(n-1)..A014138(n-1)] of permutation A089867/A089868, i.e., the number of n-node binary trees fixed by the corresponding automorphism(s).

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

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).

Tian-Xiao He and Renzo Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), no. 12, 3962-3974. [N. J. A. Sloane, Nov 26 2011]

Ângela Mestre and José Agapito, A Family of Riordan Group Automorphisms, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.

FORMULA

CONV(A000045, A000108).

G.f.: (1-(1-4x)^(1/2))/(2(1-x-x^2)). The generating function for the convolution of Catalan and Fibonacci numbers is simply the generating functions of the Catalan and Fibonacci numbers multiplied together. - Molly Leonard (maleonard1(AT)stthomas.edu), Aug 04 2006

For n>1, a(n) = a(n-1) + a(n-2) + A000108(n-1). - Gerald McGarvey, Sep 19 2008

Conjecture: n*a(n) + (-5*n+6)*a(n-1) + 3*(n-2)*a(n-2) + 2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jul 09 2013

a(n) = A139375(n,1) for n > 0. - Reinhard Zumkeller, Aug 28 2013

a(n) ~ 2^(2*n + 2) / (11*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 10 2018

MATHEMATICA

CoefficientList[Series[(1-(1-4x)^(1/2))/(2(1-x-x^2)), {x, 0, 30}], x] (* Harvey P. Dale, Apr 05 2011 *)

PROG

(MIT/GNU Scheme) (define (A090826 n) (convolve A000045 A000108 n))

(define (convolve fun1 fun2 upto_n) (let loop ((i 0) (j upto_n)) (if (> i upto_n) 0 (+ (* (fun1 i) (fun2 j)) (loop (+ i 1) (- j 1))))))

(Haskell)

import Data.List (inits)

a090826 n = a090826_list !! n