A003240 Number of partially achiral rooted trees.

1, 1, 2, 4, 8, 16, 31, 62, 120, 236, 454, 884, 1697, 3275, 6266, 12020, 22935, 43788, 83325, 158516, 300914, 570794, 1081157, 2045934, 3867617, 7304149, 13783221, 25984936, 48956715, 92155376, 173376484, 325919786, 612378787, 1149777034

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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..3760 (terms 1..70 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)
• Index entries for sequences related to rooted trees
• 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=0,100,t(i)*y^(2*i)); p=subst(y*Ty2/(y-Ty2),y,y+y*O(y^n));p=Pol(p,y); r=subst(Ty2*(y+p+(p^2-subst(p,y,y^2))/(2*y))/y^2,y,x+x*O(x^n)); for(I=1,n-2,print1(polcoeff(r,i)","))} \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008

Formula

a(n) ~ c * d^n * n, where d = 1.8332964415228533737988849634129366404833316666328290543862325494628120733... is the root of the equation Sum_{k>=1} A000081(k) / d^(2*k-1) = 1 and c = 0.030410107348865811204534352170117292921782094079168428605205142049899... - Vaclav Kotesovec, Dec 13 2020

Extensions

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