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Number of partially achiral trees with n nodes.
(Formerly M0760)
1

%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