A003237 Number of partially achiral planted trees with n nodes.
0, 0, 1, 1, 2, 3, 6, 10, 19, 33, 62, 110, 204, 366, 677, 1223, 2254, 4089, 7526, 13692, 25171, 45882, 84291, 153860, 282509, 516192, 947469, 1732477, 3179083, 5816301, 10670751, 19531034, 35826689, 65596323, 120312363, 220340374, 404096665, 740212002, 1357426934
Offset: 0
Keywords
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..3770 (terms 0..1000 from Vincenzo Librandi)
- F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Index entries for sequences related to rooted trees
- Index entries for sequences related to trees
Programs
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Maple
G := subs(x=x^2,G000081); x*G/(x-G); # second Maple program: 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; unapply(add(b(k)*x^k, k=1..n),x) end: a:= n-> coeff(series(x* B(floor(n/2))(x^2)/ (x-B(floor(n/2))(x^2)), x=0, n+2),x,n): seq(a(n), n=0..38); # Alois P. Heinz, Aug 21 2008
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Mathematica
max = 38; a81[n_] := a81[n] = If[n <= 1, n, Sum[Sum[d*a81[d], {d, Divisors[j]}]*a81[n-j], {j, 1, n-1}]/(n-1)]; G81[x_] = Sum[a81[k]*x^k, {k, 0, max}]; G[x_] = G81[x^2]; A[x_] = x*(G[x]/(x-G[x])); CoefficientList[Series[A[x], {x, 0, max}], x] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)
Formula
G.f.: A(x) = x*G(x)/(x-G(x)), where G(x) = G000081(x^2), G000081(x) = x+x^2+2*x^3+ ... being the g.f. for A000081.
a(n) ~ c * d^n, where d = 1.8332964415228533737988849634129366404833316666328290543862325494628120733... is the root of the equation Sum_{k>=1} A000081(k) / d^(2*k-1) = 1 and c = 0.1345213691789841963849894554233223547113840356469443704501548999022472... - Vaclav Kotesovec, Dec 13 2020
Extensions
Entry revised Mar 25 2004
Comments