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A000671 Boron trees with n nodes = n-node rooted trees with deg <=3 at root and out-degree <=2 elsewhere.
(Formerly M1083 N0411)
2
0, 1, 1, 2, 4, 7, 14, 29, 60, 127, 275, 598, 1320, 2936, 6584, 14858, 33744, 76999, 176557, 406456, 939241, 2177573, 5064150, 11809632, 27610937, 64705623, 151966597, 357623905, 843176524, 1991439229, 4711115672, 11162025770 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The subsequence of primes begins: 2, 7, 29, 127, 176557, 2177573, 151966597.

REFERENCES

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).

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

R. C. Read, personal communication.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

Index entries for sequences related to rooted trees

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Index entries for sequences related to trees

FORMULA

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)).

MAPLE

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);

MATHEMATICA

max = 31; G001190[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = c[2] = c[3] = 1; solc = SolveAlways[ Normal[ Series[ G001190[x] - (x + (1/2)*(G001190[x]^2 + G001190[x^2])), {x, 0, max}]] == 0, x]; f[x_] := G001190[x]/x; g[x_] := Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = a[2] = 1; sola = SolveAlways[ Normal[ Series[ g[x] - (x/3!)*(f[x]^3 + 3*f[x^2]*f[x] + 2*f[x^3]) /. solc[[1]], {x, 0, max}]] == 0, x]; Table[a[n], {n, 0, max}] /. sola[[1]] (* Jean-Fran├žois Alcover, May 29 2012, from first g.f. *)

CROSSREFS

Sequence in context: A119342 A119268 A002989 * A199888 A157133 A202850

Adjacent sequences:  A000668 A000669 A000670 * A000672 A000673 A000674

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 20 21:08 EDT 2017. Contains 292293 sequences.