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A000106 2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.
(Formerly M1415 N0553)
14
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)
OFFSET
2,2
REFERENCES
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).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 2..1000 (terms n = 2..200 from T. D. Noe)
Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
FORMULA
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
In the asymptotics above the constant c = 2 * A187770. - Vladimir Reshetnikov, Aug 13 2016
MAPLE
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
MATHEMATICA
<<NumericalDifferentialEquationAnalysis`; btc = ButcherTreeCount[max = 30]; Flatten[ Table[ ListConvolve[t=Take[btc, n], t], {n, 1, max}]] (* Jean-François Alcover, Nov 02 2011 *)
b[n_] := b[n] = If[n <= 1, n, Sum[k*b[k]*s[n-1, k], {k, 1, n-1}]/(n-1)]; s[n_, k_] := s[n, k] = Sum[b[n+1-j*k], {j, 1, Quotient[n, k]}]; B[n_] := B[n] = Sum[b[k]*x^k, {k, 1, n}]; a[n_] := SeriesCoefficient[B[n-1]^2, {x, 0, n}]; Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Dec 01 2016, after Alois P. Heinz *)
PROG
(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
CROSSREFS
Column d=1 of A335362.
Column 2 of A339067.
Cf. A000081, A000242, A000300, A000343, A000395, A027852 (forests of 2 rooted trees).
Sequence in context: A118649 A033482 A054341 * A076883 A140832 A326793
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from Christian G. Bower, Nov 15 1999
STATUS
approved

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Last modified March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)