0,4

A. Cayley, On the analytical forms called trees, with application to the theory of chemical combinations, Reports British Assoc. Advance. Sci. 45 (1875), 257-305 = Math. Papers, Vol. 9, 427-460 (see p. 450).

R. C. Read, personal communication.

S. J. Cyvin et al., Enumeration of constitutional isomers of polyenes, J. Molec. Structure (Theochem), 357 (1995), 255-261.

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for sequences related to rooted trees

Index entries for sequences related to trees

G.f.: A(x) = x*(1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)), where f = G001190(x)/x, G001190 = g.f. for A001190.

A000671(n) = A001190(n) + A036657(n) + A036658(n).

Another g.f.: let B0(x) = 1+x, G036656(x) = g.f. for A036656, G036657(x) = g.f. for A036657.

Then g.f. = x*(cycle_index(S3, B0)+cycle_index(S3, G036656)+cycle_index(S3, G036657)+cycle_index(S2, B0)*(G036656+G036657)+cycle_index(S2, G036656)*(G036657+B0)+cycle_index(S2, G036657)*(B0+G036656)+B0*G036656*G036657), where cycle_index(Sk, f) means apply the cycle index for the symmetric group S_k to f(x).

E.g. cycle_index(S2, f) = (1/2!)*(f^2+subs(x=x^2, f), cycle_index(S3, f) = (1/3!)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)).

N := 40: t1 := G001190/x: G000671 := series(x*(1/3!)*(t1^3+3*subs(x=x^2, t1)*t1+2*subs(x=x^3, t1)), x, N); A000671 := n->coeff(G000671, x, n);

CI2 := proc(f) (1/2)*(f^2+subs(x=x^2, f)); end; CI3 := proc(f) (1/6)*(f^3+3*subs(x=x^2, f)*f+2*subs(x=x^3, f)); end;

N := 40: B0 := series(1 + x, x, N): G000671 := series(x*(CI3(B0) + CI3(G036656) + CI3(G036657) + CI2(B0)*(G036656 + G036657) + CI2(G036656)*(G036657 + B0) + CI2(G036657)*(B0 + G036656) + B0*G036656*G036657), x, N); A036658 := n->coeff(G036658, x, n);

Sequence in context: A119342 A119268 A002989 this_sequence A120262 A013326 A074663

Adjacent sequences: A000668 A000669 A000670 this_sequence A000672 A000673 A000674

nonn,easy,nice

njas