Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I M0760 #37 Apr 13 2022 13:25:16
%S 1,1,1,2,3,6,9,19,30,61,99,198,333,650,1115,2143,3743,7101,12553,
%T 23605,42115,78670,141284,262679,474083,878386,1591038,2940512,
%U 5340712,9852201,17930619,33031498,60209609,110801271,202208576,371820314
%N Number of partially achiral trees with n nodes.
%C 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.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A003243/b003243.txt">Table of n, a(n) for n = 1..3770</a> (terms 1..73 from Herman Jamke)
%H F. Harary and R. W. Robinson, <a href="http://gdz.sub.uni-goettingen.de/en/dms/loader/img/?PID=GDZPPN002191393">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335.
%H F. Harary and R. W. Robinson, <a href="/A002995/a002995_1.pdf">The number of achiral trees</a>, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F 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
%o (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
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_
%E More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Feb 26 2008