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A124497
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Number of 1-2-3 trees with n edges and with thinning limbs. A 1-2-3 tree is an ordered tree with vertices of outdegree at most 3. A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.
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4
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1, 1, 2, 4, 9, 20, 48, 116, 288, 724, 1849, 4768, 12423, 32628, 86342, 229952, 616042, 1659012, 4489101, 12199521, 33284546, 91140797, 250396629, 690043032, 1907022197, 5284167884, 14677681554, 40862469713, 114001697975
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OFFSET
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0,3
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LINKS
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Table of n, a(n) for n=0..28.
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FORMULA
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G.f.=H*T(H^2*z^3), where T=2/sqrt(3*x)*sin((1/3)*arcsin(sqrt(27*x/4))) (solution of T=1+zT^3, T(0)=1), H=C(z^2/(1-z))/(1-z) and C(x)=[1-sqrt(1-4x)]/(2x) is the Catalan function. More generally, if M[k](z) is the g.f. of the 1-2-...-k trees with thinning limbs and C[k](z)=1+z*{C[k](z)}^k is the g.f. of the k-ary trees, then M[k](z)=M[k-1](z)C[k](M[k-1]^(k-1)*z^k).
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MAPLE
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C:=x->(1-sqrt(1-4*x))/2/x: T:=x->2/sqrt(3*x)*sin((1/3)*arcsin(sqrt(27*x/4))): M2:=C(z^2/(1-z))/(1-z): G:=M2*T(M2^2*z^3): Gser:=series(G, z=0, 40): seq(coeff(Gser, z, n), n=0..33);
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CROSSREFS
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Cf. A090344, A124344.
Sequence in context: A145549 A145550 A000081 * A093637 A068051 A032289
Adjacent sequences: A124494 A124495 A124496 * A124498 A124499 A124500
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch and Louis Shapiro (deutsch(AT)duke.poly.edu, lshapiro(AT)Howard.edu), Nov 04 2006
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STATUS
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approved
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