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 A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes. (Formerly M4757) 2
 10, 840, 257040, 137260200, 118273755600, 154712104747200, 292311804557572800, 766931112143320924800, 2706462791802644002128000, 12512595130808078973370704000, 74130965352250071944327288640000, 552334353713465817349513210512960000, 5092566798555894395129552704613028960000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 3,1 COMMENTS R. C. Read incorrectly has a(7) = 118257539400 and a(8) = 154678050727200 which he calculated by hand. - Sean A. Irvine, Jun 27 2017 REFERENCES R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Andrew Howroyd, Table of n, a(n) for n = 3..50 R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence) FORMULA a(n) = A246599(n) + Sum_{k=1..n-1} binomial(2*n-1,2*k-1)*A246599(k)*a(n-k). - Andrew Howroyd, May 22 2018 MATHEMATICA (* b stands for A001501 *) b[n_] := n!^2 Sum[2^(2k-n) 3^(k-n) (3(n-k))!  HypergeometricPFQ[{k-n, k-n}, {3(k-n)/2, 1/2 + 3(k-n)/2}, -9/2]/(k! (n-k)!^2), {k, 0, n}]/6^n; (* c stands for A246599 *) c[n_] := c[n] = Binomial[2n-1, n] b[n] - Sum[ Binomial[2n-1, 2k] Binomial[2k, k] b[k] c[n-k], {k, 1, n-1}]; a[n_] := a[n] = c[n] + Sum[Binomial[2n-1, 2k-1] c[k] a[n-k], {k, 1, n-1}]; Table[a[n], {n, 3, 20}] (* Jean-François Alcover, Jul 07 2018, after Andrew Howroyd *) PROG (PARI) \\ here b(n) is A001501 b(n) = {n!^2 * sum(j=0, n, sum(i=0, n-j, my(k=n-i-j); (j + 3*k)! / (3^i * 36^k * i! * k!^2)) / (j! * (-2)^j))} seq(n)={my(v=vector(n, n, b(n)*binomial(2*n-1, n)), u=vector(n), s=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, 2*binomial(2*n-1, 2*k)*v[k]*u[n-k]); s[n]=u[n] + sum(k=3, n-3, binomial(2*n-1, 2*k-1)*u[k]*s[n-k])); s[3..n]} \\ Andrew Howroyd, May 22 2018 CROSSREFS Cf. A001501, A002829, A004109, A246599. Sequence in context: A013436 A013437 A246599 * A203533 A015033 A126677 Adjacent sequences:  A006711 A006712 A006713 * A006715 A006716 A006717 KEYWORD nonn AUTHOR EXTENSIONS a(7)-a(8) corrected and a(9)-a(12) computed with nauty by Sean A. Irvine, Jun 27 2017 Terms a(13) and beyond from Andrew Howroyd, May 22 2018 STATUS approved

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Last modified January 20 05:27 EST 2020. Contains 331067 sequences. (Running on oeis4.)