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Number of symmetric short bushes with n edges.
1

%I #12 Jul 24 2022 12:02:11

%S 1,0,1,1,1,2,3,4,7,10,17,25,43,64,111,167,291,442,773,1183,2075,3196,

%T 5619,8702,15329,23852,42085,65755,116181,182186,322287,507020,897859,

%U 1416594,2510901,3971887,7045915,11171924,19832947,31514404,55982893

%N Number of symmetric short bushes with n edges.

%C Or number of ordered trees with n edges, no vertices of outdegree 1 and which are symmetrical with respect to the vertical axis passing through the root.

%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.

%F G.f.: [(1-z)(1+z^2)-(1+z)sqrt(1-2z^2-3z^4)]/[2z(z^3+z^2+z-1)].

%F a(n) has antiparity of A007814(n+1), i.e. a(n) mod 2 = A035263(n+1). - _Ralf Stephan_, Feb 21 2004

%F D-finite with recurrence (n+1)*a(n) -2*a(n-1) +4*(-n+1)*a(n-2) +2*(-n+2)*a(n-3) +4*a(n-4) +2*(2*n-3)*a(n-5) +4*(2*n-11)*a(n-6) +6*(n-6)*a(n-7) +3*(n-7)*a(n-8)=0. - _R. J. Mathar_, Jul 24 2022

%K nonn

%O 0,6

%A _Emeric Deutsch_, May 26 2003