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

%I A246599 #25 Feb 17 2024 04:02:35
%S A246599 10,840,257040,137214000,118248530400,154686980448000,
%T A246599 292276881344448000,766864651478365440000,2706292794907249067520000,
%U A246599 12512021073989410699165440000,74128448237031250090060032000000,552320243355746711191770103680000000,5092467146398443040845772685937408000000
%N A246599 Number of connected trivalent bipartite labeled graphs with 2n labeled nodes.
%C A246599 R. C. Read incorrectly has a(7) = 118237555800 and a(8) = 154652926428000 which he calculated by hand.
%D A246599 R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%H A246599 Andrew Howroyd, <a href="/A246599/b246599.txt">Table of n, a(n) for n = 3..50</a>
%H A246599 R. C. Read, <a href="/A002831/a002831.pdf">Letter to N. J. A. Sloane, Feb 04 1971</a> (gives initial terms of this sequence)
%F A246599 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
%F A246599 a(n) ~ 3^(n + 1/2) * n^(3*n) / (sqrt(2) * exp(3*n + 2)). - _Vaclav Kotesovec_, Feb 17 2024
%t A246599 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;
%t A246599 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}];
%t A246599 Table[a[n], {n, 3, 20}] (* _Jean-François Alcover_, Jul 07 2018, after _Andrew Howroyd_ *)
%o A246599 (PARI) \\ here b(n) is A001501
%o A246599 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))}
%o A246599 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
%Y A246599 Cf. A001501, A002829, A004109, A006714.
%K A246599 nonn
%O A246599 3,1
%A A246599 _N. J. A. Sloane_, Sep 08 2014
%E A246599 a(7)-a(8) corrected and a(9)-a(12) computed with nauty by _Sean A. Irvine_, Jun 27 2017
%E A246599 Terms a(13) and beyond from _Andrew Howroyd_, May 22 2018