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
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