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A090826 Convolution of Catalan and Fibonacci numbers. 6
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 (list; graph; refs; listen; history; text; internal format)
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).

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

REFERENCES

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

LINKS

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

S. B. Ekhad, 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

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

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 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

a090826_list = map (sum . zipWith (*) a000045_list . reverse) $

                   tail $ inits a000108_list

-- Reinhard Zumkeller, Aug 28 2013

CROSSREFS

Cf. Catalan numbers: A000108, Fibonacci numbers: A000045.

Sequence in context: A160999 A014329 A045633 * A132441 A000840 A232215

Adjacent sequences:  A090823 A090824 A090825 * A090827 A090828 A090829

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Dec 20 2003

STATUS

approved

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Last modified November 20 10:41 EST 2017. Contains 294963 sequences.