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A000106
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2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.
(Formerly M1415 N0553)
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8
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1, 2, 5, 12, 30, 74, 188, 478, 1235, 3214, 8450, 22370, 59676, 160140, 432237, 1172436, 3194870, 8741442, 24007045, 66154654, 182864692, 506909562, 1408854940, 3925075510, 10959698606, 30665337738, 85967279447, 241433975446, 679192039401, 1913681367936, 5399924120339
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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2,2
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REFERENCES
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J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 and Alois P. Heinz, Table of n, a(n) for n = 2..1000 (terms n = 2..200 from T. D. Noe)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 385
Index entries for sequences related to rooted trees
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FORMULA
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Self-convolution of rooted trees A000081.
a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.87984802514205060808180678... . - Vaclav Kotesovec, Sep 11 2014
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MAPLE
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b:= proc(n) option remember; if n<=1 then n else add(k*b(k)* s(n-1, k), k=1..n-1)/(n-1) fi end: s:= proc(n, k) option remember; add(b(n+1-j*k), j=1..iquo(n, k)) end: B:= proc(n) option remember; add(b(k)*x^k, k=1..n) end: a:= n-> coeff(series(B(n-1)^2, x=0, n+1), x, n): seq(a(n), n=2..35); # Alois P. Heinz, Aug 21 2008
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MATHEMATICA
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<<NumericalDifferentialEquationAnalysis`; btc = ButcherTreeCount[max = 30]; Flatten[ Table[ ListConvolve[t=Take[btc, n], t], {n, 1, max}]] (* Jean-François Alcover, Nov 02 2011 *)
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PROG
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(Haskell)
a000106 n = a000106_list !! (n-2)
a000106_list = drop 2 $ conv a000081_list [] where
conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'
where ws' = v : ws
-- Reinhard Zumkeller, Jun 17 2013
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CROSSREFS
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Cf. A000081, A000242, A000300, A000343, A000395.
Sequence in context: A118649 A033482 A054341 * A076883 A140832 A026580
Adjacent sequences: A000103 A000104 A000105 * A000107 A000108 A000109
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Christian G. Bower, Nov 15 1999
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STATUS
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approved
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