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A090826
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Convolution of Catalan and Fibonacci numbers.
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6
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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
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0,3
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COMMENTS
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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).
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LINKS
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FORMULA
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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
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
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MATHEMATICA
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CoefficientList[Series[(1-(1-4x)^(1/2))/(2(1-x-x^2)), {x, 0, 30}], x] (* Harvey P. Dale, Apr 05 2011 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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