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%I M3871 N1586 #41 May 03 2023 09:14:20
%S 1,1,5,16,86,448,3580,34981,448628,6854130,121173330,2403140605,
%T 52655943500,1260724587515,32726520985365,915263580719998,
%U 27432853858637678,877211481667946811,29807483816421710806,1072542780403547030073,40739888428757581326987
%N Number of 3-edge-colored trivalent graphs with 2n nodes.
%D R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A002830/b002830.txt">Table of n, a(n) for n = 0..200</a>
%H Nicholas Dub, <a href="https://tel.archives-ouvertes.fr/tel-03641958/">Enumeration of triangulations modulo symmetries and of rooted triangulations counted by their number of (d - 2)-simplices in dimension d ≥ 2</a>, tel-03641958 [cs.OH], Université Paris-Nord - Paris XIII, 2021.
%H Sean A. Irvine, <a href="/A002830/a002830.gif">Illustration of initial terms</a>
%H 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 G.f.: exp(Sum_{k >= 1} F(x^k) / k) where F(x) is the g.f. for A002831. - _Sean A. Irvine_, Sep 09 2014
%t permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t k; s += t]; s!/m];
%t b[k_, q_] := If[OddQ[q], If[OddQ[k], 0, j = k/2; q^j (2 j)!/(j! 2^j)], Sum[ Binomial[k, 2 j] q^j (2 j)!/(j! 2^j), {j, 0, Quotient[k, 2]}]];
%t pm[v_] := Module[{p = Total[x^v]}, Product[b[Coefficient[p, x, i], i], {i, 1, Exponent[p, x]}]];
%t a[n_] := Module[{s = 0}, Do[s += permcount[p] pm[p]^3, {p, IntegerPartitions[2 n]}]; s/(2 n)!];
%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 30}] (* _Jean-François Alcover_, Jul 02 2018, after _Andrew Howroyd_ *)
%o (PARI)
%o b(k,r) = {if(k%2, if(r%2, 0, my(j=r/2); k^j*(2*j)!/(j!*2^j)), sum(j=0, r\2, binomial(r, 2*j)*k^j*(2*j)!/(j!*2^j)))}
%o g(n,k)={sum(r=0, n\k, x^(k*r)*b(k,r)^3/(r!*k^r)) + O(x*x^n)}
%o seq(n)={Vec(substpol(prod(k=1, 2*n, g(2*n,k)), x^2, x))} \\ _Andrew Howroyd_, Dec 14 2017; updated May 02 2023
%Y Cf. A002831, A006712, A006713.
%K nonn
%O 0,3
%A _N. J. A. Sloane_
%E a(7)-a(8) from _Sean A. Irvine_, Sep 08 2014
%E Terms a(9) and beyond from _Andrew Howroyd_, Dec 14 2017
%E a(0)=1 prepended by _Andrew Howroyd_, May 02 2023
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