A000106 - OEIS

%I M1415 N0553 #65 Nov 03 2023 16:18:54

%S 1,2,5,12,30,74,188,478,1235,3214,8450,22370,59676,160140,432237,

%T 1172436,3194870,8741442,24007045,66154654,182864692,506909562,

%U 1408854940,3925075510,10959698606,30665337738,85967279447,241433975446,679192039401,1913681367936,5399924120339

%N 2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 150.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A000106/b000106.txt">Table of n, a(n) for n = 2..1000</a> (terms n = 2..200 from T. D. Noe)

%H Vsevolod Gubarev, <a href="https://arxiv.org/abs/1811.08219">Rota-Baxter operators on a sum of fields</a>, arXiv:1811.08219 [math.RA], 2018.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=385">Encyclopedia of Combinatorial Structures 385</a>

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F Self-convolution of rooted trees A000081.

%F a(n) ~ c * d^n / n^(3/2), where d = A051491 = 2.9557652856519949747148..., c = 0.87984802514205060808180678... . - _Vaclav Kotesovec_, Sep 11 2014

%F In the asymptotics above the constant c = 2 * A187770. - _Vladimir Reshetnikov_, Aug 13 2016

%p 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

%t <<NumericalDifferentialEquationAnalysis`; btc = ButcherTreeCount[max = 30]; Flatten[ Table[ ListConvolve[t=Take[btc, n], t], {n, 1, max}]] (* _Jean-François Alcover_, Nov 02 2011 *)

%t 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_ *)

%o (Haskell)

%o a000106 n = a000106_list !! (n-2)

%o a000106_list = drop 2 $ conv a000081_list [] where

%o    conv (v:vs) ws = (sum $ zipWith (*) ws' $ reverse ws') : conv vs ws'

%o                     where ws' = v : ws

%o -- _Reinhard Zumkeller_, Jun 17 2013

%Y Column d=1 of A335362.

%Y Column 2 of A339067.

%Y Cf. A000081, A000242, A000300, A000343, A000395, A027852 (forests of 2 rooted trees).

%K nonn,nice,easy

%O 2,2

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_, Nov 15 1999