A286757 Number of labeled connected rooted trivalent graphs with 2n nodes.

0, 4, 120, 33600, 18471600, 18386121600, 30231607606200, 76388992266787200, 281063897503929540000, 1444102677105174358272000, 10020068498645397815029407000, 91355440119583548608158042584000, 1069762020017605579789451640683370000

Offset: 1

Author

Sean A. Irvine, May 13 2017

Keywords

• nonn

Comments

A006607 gives values matching Table 1 (p. 342) of Wormald. However, the values in the table for n > 4 do not appear to match formulas given for generating the table.

References

• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.

Links

• N. C. Wormald, Triangles in labeled cubic graphs, pp. 337-345 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.

Crossrefs

Cf. A002829, A006607.

Formula

Let b(0)=b(1)=0, b(n) = 2*binomial(2*n, 2)*b(n-1) + 12*binomial(2*n, 4)*b(n-2) + 6*binomial(2*n, 3)*A002829(n-1) + 60*binomial(2*n, 5)*A002829(n-2) + 1260*binomial(2*n, 7)*A002829(n-3). a(n)=b(n) except a(2)=4.

Let Q(x) be an e.g.f. for A002829: Q(x) = 1 + (1/4!)*x^4 + (70/6!)*x^6 + (19355/8!)*x^8 + ...; then A(x), the e.g.f. for this sequence, satisfies (2 - 2*x^2 - x^4) * (A(x) - (1/6)*x^4) = (2*x^3 + x^5 + (1/2)*x^7) * Q'(x) where Q'(x) is the derivative of Q(x) with respect to x.