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A003214
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Number of binary forests with n nodes.
(Formerly M0775)
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5
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1, 1, 2, 3, 6, 10, 20, 37, 76, 152, 320, 672, 1454, 3154, 6959, 15439, 34608, 77988, 176985, 403510, 924683, 2127335, 4913452, 11385955, 26468231, 61700232, 144206269, 337837221, 793213550, 1866181155, 4398867672, 10387045476, 24567374217, 58196129468, 138056734916
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OFFSET
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0,3
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COMMENTS
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From Piet Hut, Nov 07, 2003: "Number of ways to place n stars in stable hierarchical multiple star systems (where each stable multiple is a binary tree: around its center of mass two multiple star systems revolve, each of which can be a singleton or a nontrivial multiple star system).
"For example, a(1) = 1 : *; a(2) = 2 : (**), * *; a(3) = 3 : ((**)*), (**) *, * * *; a(4) = 6 : (((**)*)*), ((**)(**)), ((**)*) *, (**) (**), (**) * *, * * * * ."
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REFERENCES
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L. F. Meyers, Corrections and additions to Tree Representations in Linguistics. Report 3, 1966, p. 138. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
L. F. Meyers and W. S.-Y. Wang, Tree Representations in Linguistics. Report 3, 1963, pp. 107-108. Project on Linguistic Analysis, Ohio State University Research Foundation, Columbus, Ohio.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Piet Hut, Home Page
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FORMULA
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Euler transform of A001190. - Michael Somos, Nov 10 2003
G.f.: exp( sum G001190(x^i)/i, i=1..infinity ), where G001190 = g.f. for A001190.
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MATHEMATICA
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max = 34; c[0] = 0; g[x_] = Sum[c[k]*x^k, {k, 0, max}]; eq[0] = Rest[ Thread[ CoefficientList[ (-2*x + 2*g[x] - g[x]^2 - g[x^2])/2, x] == 0]]; s[1] = First[ Solve[ First[eq[0]], c[1]]]; Do[ eq[k-1] = Rest[ eq[k-2]] /. s[k-1]; s[k] = First[ Solve[ First[eq[k-1]], c[k]]], {k, 2, max}]; sol = Flatten[ Table[ s[k], {k, 1*max}], 1]; f[x_] = Exp[ Sum[ g[x^i]/i, {i, 1, max}]] /. sol; CoefficientList[ Series[ f[x], {x, 0, max}], x](* From Jean-François Alcover, Nov 18 2011 *)
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CROSSREFS
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Cf. A001190.
Sequence in context: A045690 A007148 A093371 * A123423 A005195 A052843
Adjacent sequences: A003211 A003212 A003213 * A003215 A003216 A003217
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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