A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.

10, 840, 257040, 137214000, 118248530400, 154686980448000, 292276881344448000, 766864651478365440000, 2706292794907249067520000, 12512021073989410699165440000, 74128448237031250090060032000000, 552320243355746711191770103680000000, 5092467146398443040845772685937408000000

Offset

3,1

Comments

R. C. Read incorrectly has a(7) = 118237555800 and a(8) = 154652926428000 which he calculated by hand.

References

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

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) = binomial(2*n-1, n)*A001501(n) - Sum_{k=1..n-1} binomial(2*n-1, 2*k) * binomial(2*k, k) * A001501(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[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;
a[n_] := a[n] = Binomial[2n-1, n] b[n] - Sum[Binomial[2n-1, 2k] Binomial[2 k, k] b[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, n)), u=vector(n)); for(n=1, #u, u[n]=v[n] - sum(k=3, n-3, binomial(2*n-1, 2*k)*v[k]*u[n-k])); u[3..n]/2} \\ Andrew Howroyd, May 22 2018

Crossrefs

Cf. A001501, A002829, A004109, A006714.

Sequence in context: A013434 A013436 A013437 * A006714 A203533 A015033

Adjacent sequences: A246596 A246597 A246598 * A246600 A246601 A246602

Keyword

nonn

Author

N. J. A. Sloane, Sep 08 2014

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