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A228365 Inverse binomial transform of the Galois numbers G_(n)^{(3)} (A006117).

%I A228365 #20 May 05 2025 08:32:45
%S A228365 1,1,3,15,129,1833,43347,1705623,112931553,12639552945,2413134909507,
%T A228365 788041911546303,442817851480763169,428369525248261655193,
%U A228365 716160018275094098267859,2067365673240491189928496263,10333740296321620864171488891201,89302459853776656431139970491341025
%N A228365 Inverse binomial transform of the Galois numbers G_(n)^{(3)} (A006117).
%C A228365 Analog of the inverse binomial transform of G_(n)^{(q)} with q=2, A135922.
%C A228365 A 2-multigraph is a labeled graph with no loops but with up to 2 edges joining any pair of vertices.  a(n) is the number of 2-multigraphs on [n] such that no path of length two has vertices i,j,k (in that order) with i<j<k.  See Stanley-Locke link below. - _Geoffrey Critzer_, May 05 2025
%H A228365 Alois P. Heinz, <a href="/A228365/b228365.txt">Table of n, a(n) for n = 0..91</a>
%H A228365 R. P. Stanley and S. C. Locke, <a href="https://www.jstor.org/stable/2589063">Graphs without increasing paths: Solution to Problem 10572</a>, The American Mathematical Monthly, 106(2) (1999), 168.
%F A228365 a(n) ~ c * 3^(n^2/4), where c = EllipticTheta[3,0,1/3]/QPochhammer[1/3,1/3] = 3.019783845699... if n is even and c = EllipticTheta[2,0,1/3]/QPochhammer[1/3,1/3] = 3.01826904637117... if n is odd. - _Vaclav Kotesovec_, Aug 23 2013
%p A228365 b:= proc(n) option remember; add(mul(
%p A228365       (3^(i+k)-1)/(3^i-1), i=1..n-k), k=0..n)
%p A228365     end:
%p A228365 a:= proc(n) option remember;
%p A228365       add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
%p A228365     end:
%p A228365 seq(a(n), n=0..19);  # _Alois P. Heinz_, Sep 24 2019
%t A228365 Table[SeriesCoefficient[Sum[x^n/Product[(1-(3^k-1)*x),{k,0,n}],{n,0,nn}],{x,0,nn}] ,{nn,0,20}] (* _Vaclav Kotesovec_, Aug 23 2013 *)
%Y A228365 Cf. A135922, A006117, A006116.
%K A228365 nonn
%O A228365 0,3
%A A228365 _R. J. Mathar_, Aug 21 2013