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A006470
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Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.
(Formerly M2075)
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4
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2, 15, 60, 175, 420, 882, 1680, 2970, 4950, 7865, 12012, 17745, 25480, 35700, 48960, 65892, 87210, 113715, 146300, 185955, 233772, 290950, 358800, 438750, 532350, 641277, 767340, 912485, 1078800, 1268520, 1484032, 1727880, 2002770, 2311575, 2657340, 3043287, 3472820, 3949530, 4477200, 5059810, 5701542, 6406785, 7180140
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OFFSET
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1,1
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COMMENTS
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a(n) is the number of ordered rooted trees with n+3 non-root nodes that have 3 leaves; see A108838. - Joerg Arndt, Aug 18 2014
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REFERENCES
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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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FORMULA
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a(n) = (n+1)*binomial(n+3, 4).
a(n) = binomial(n+2, 2)*binomial(n+4, 3)/2;
G.f.: x*(2+3*x)/(1-x)^6. (End)
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).
a(n) = n*(n+1)^2*(n+2)*(n+3)/24. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 16*log(2) + 5/3. - Amiram Eldar, Jan 28 2022
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MAPLE
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MATHEMATICA
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Table[n (n + 1)^2 (n + 2) (n + 3) / 24, {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
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PROG
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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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EXTENSIONS
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STATUS
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approved
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