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 A125189 Number of symmetric bushes with n edges. I.e., number of ordered trees with n edges, no non-root vertices of outdegree 1 and symmetrical with respect to the vertical axis passing through the root. 1
 1, 1, 1, 2, 2, 3, 5, 7, 11, 17, 27, 42, 68, 107, 175, 278, 458, 733, 1215, 1956, 3258, 5271, 8815, 14321, 24031, 39181, 65937, 107840, 181936, 298367, 504473, 829307, 1404879, 2314453, 3927495, 6482788, 11017802, 18217839, 31004871, 51347351, 87497297 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Robert Israel, Table of n, a(n) for n = 0..4206 J.-L. Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016). R. Donaghey and L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301. FORMULA a(n) = A082958(n) + A082958(n-1) for n >= 1 (every symmetric bush with n edges consists of the symmetric short bushes with n edges and the symmetric short bushes with n-1 edges hanging on an edge emanating from the root). G.f.: (1+z)*((1-z)*(1+z^2)-(1+z)*sqrt(1-2*z^2-3*z^4))/(2*z*(z^3+z^2+z-1)). Conjecture: (n+1)*a(n) -3*a(n-1) +(-4*n+5)*a(n-2) +(-2*n+7)*a(n-3) +3*a(n-4) +(4*n-5)*a(n-5) +(8*n-49)*a(n-6) +3*(2*n-13)*a(n-7) +3*(n-8)*a(n-8)=0. - R. J. Mathar, Jun 08 2016 The conjecture follows from the differential equation 3*z^7 + z^6 + 3*z^5 + 5*z^4 + z^3 + 3*z^2 + z - 1 + (3*z^7 - z^6 + 15*z^5 + 3*z^4 + z^3 - 3*z^2 - 3*z + 1)*g(z) + (3*z^9 + 6*z^8 + 8*z^7 + 4*z^6 - 2*z^4 - 4*z^3 + z)*g'(z) = 0 satisfied by the g.f.. - Robert Israel, Nov 21 2017 MAPLE G:=(1+z)*((1-z)*(1+z^2)-(1+z)*sqrt(1-2*z^2-3*z^4))/(2*z*(z^3+z^2+z-1)): Gser:=series(G, z=0, 50): seq(coeff(Gser, z, n), n=0..45); CROSSREFS Sequence in context: A066889 A214040 A077419 * A226498 A196375 A300440 Adjacent sequences: A125186 A125187 A125188 * A125190 A125191 A125192 KEYWORD nonn AUTHOR Emeric Deutsch, Dec 20 2006 STATUS approved

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