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%I #11 Jan 04 2021 23:04:05
%S 0,4,120,33600,18471600,18386121600,30231607606200,76388992266787200,
%T 281063897503929540000,1444102677105174358272000,
%U 10020068498645397815029407000,91355440119583548608158042584000,1069762020017605579789451640683370000
%N Number of labeled connected rooted trivalent graphs with 2n nodes.
%C 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.
%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
%H N. C. Wormald, <a href="http://dx.doi.org/10.1007/BFb0062550">Triangles in labeled cubic graphs</a>, pp. 337-345 of Combinatorial Mathematics (Canberra, 1977), Lect. Notes Math. 686, 1978.
%F 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.
%F 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.
%Y Cf. A002829, A006607.
%K nonn
%O 1,2
%A _Sean A. Irvine_, May 13 2017
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