A003237 Number of partially achiral planted trees with n nodes. (Formerly M0766)

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

0,5

Comments

The g.f. z*(1-z**2-z**3-z**4+z**5)/(1-z-2*z**2+3*z**5) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

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

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

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

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 *)

Crossrefs

Sequence in context: A028495 A136752 A093126 * A191519 A165920 A274160

Adjacent sequences: A003234 A003235 A003236 * A003238 A003239 A003240

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Entry revised Mar 25 2004

Status

approved