A006714 Number of trivalent bipartite labeled graphs with 2n labeled nodes. (Formerly M4757)

10, 840, 257040, 137260200, 118273755600, 154712104747200, 292311804557572800, 766931112143320924800, 2706462791802644002128000, 12512595130808078973370704000, 74130965352250071944327288640000, 552334353713465817349513210512960000, 5092566798555894395129552704613028960000

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

a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n+2)). - Vaclav Kotesovec, Feb 17 2024

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

N. J. A. Sloane

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