A003243 Number of partially achiral trees with n nodes.

1, 1, 1, 2, 3, 6, 9, 19, 30, 61, 99, 198, 333, 650, 1115, 2143, 3743, 7101, 12553, 23605, 42115, 78670, 141284, 262679, 474083, 878386, 1591038, 2940512, 5340712, 9852201, 17930619, 33031498, 60209609, 110801271, 202208576, 371820314

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

The g.f. (1-z**2-2*z**3-8*z**4+7*z**5+4*z**6)/(1-z-z**2-2*z**3-6*z**4+14*z**5) was conjectured by Simon Plouffe in his 1992 dissertation, but this is incorrect.

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 = 1..3770 (terms 1..73 from Herman Jamke)
• F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
• F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
• 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 trees

Programs

PARI

t(n)=local(A=x); if(n<1, 0, for(k=1, n-1, A/=(1-x^k+x*O(x^n))^polcoeff(A, k)); polcoeff(A, n)) {n=100;Ty2=sum(i=0,n,t(i)*y^(2*i)); p=subst(y*Ty2/(y-Ty2),y,y+y*O(y^n));p=Pol(p,y);a=subst(Ty2*(y+p+(p^2-subst(p,y,y^2))/(2*y))/y^2-(p^2+subst(p,y,y^2))/(2*y^2)+Ty2,y,x+x*O(x^n)); for(i=0,n-2,print1(polcoeff(a,i)","))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008

Formula

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 = 0.123308773712306885475561730669251048497115967922743533462465528423705228... - Vaclav Kotesovec, Dec 13 2020

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008