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%I #33 Apr 12 2023 11:07:51
%S 0,1,2,5,12,31,85,248,762,2440,8064,27300,94150,329462,1166512,
%T 4170414,15031771,54559855,199236416,731434971,2697934577,9993489968,
%U 37157691565,138633745173,518851050388,1947388942885,7328186394725
%N Convolution of Catalan and Fibonacci numbers.
%C 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).
%H Reinhard Zumkeller, <a href="/A090826/b090826.txt">Table of n, a(n) for n = 0..1000</a>
%H S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%H Tian-Xiao He and Renzo Sprugnoli, <a href="https://doi.org/10.1016/j.disc.2008.11.021">Sequence characterization of Riordan arrays</a>, Discrete Math. 309 (2009), no. 12, 3962-3974. [_N. J. A. Sloane_, Nov 26 2011]
%H Ângela Mestre and José Agapito, <a href="https://www.emis.de/journals/JIS/VOL22/Agapito/mestre8.html">A Family of Riordan Group Automorphisms</a>, J. Int. Seq., Vol. 22 (2019), Article 19.8.5.
%F CONV(A000045, A000108).
%F 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
%F For n>1, a(n) = a(n-1) + a(n-2) + A000108(n-1). - _Gerald McGarvey_, Sep 19 2008
%F 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
%F a(n) = A139375(n,1) for n > 0. - _Reinhard Zumkeller_, Aug 28 2013
%F a(n) ~ 2^(2*n + 2) / (11*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 10 2018
%t CoefficientList[Series[(1-(1-4x)^(1/2))/(2(1-x-x^2)), {x,0,30}], x] (* _Harvey P. Dale_, Apr 05 2011 *)
%o (MIT Scheme) (define (A090826 n) (convolve A000045 A000108 n))
%o (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))))))
%o (Haskell)
%o import Data.List (inits)
%o a090826 n = a090826_list !! n
%o a090826_list = map (sum . zipWith (*) a000045_list . reverse) $
%o tail $ inits a000108_list
%o -- _Reinhard Zumkeller_, Aug 28 2013
%Y Cf. Catalan numbers: A000108, Fibonacci numbers: A000045.
%K nonn,easy
%O 0,3
%A _Antti Karttunen_, Dec 20 2003
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