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

1, 1, 3, 15, 129, 1833, 43347, 1705623, 112931553, 12639552945, 2413134909507, 788041911546303, 442817851480763169, 428369525248261655193, 716160018275094098267859, 2067365673240491189928496263, 10333740296321620864171488891201, 89302459853776656431139970491341025

Offset: 0

Author

R. J. Mathar, Aug 21 2013

Keywords

• nonn

Comments

Analog of the inverse binomial transform of G_(n)^{(q)} with q=2, A135922.

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 iGeoffrey Critzer, May 05 2025

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..91
• R. P. Stanley and S. C. Locke, Graphs without increasing paths: Solution to Problem 10572, The American Mathematical Monthly, 106(2) (1999), 168.

Crossrefs

Cf. A135922, A006117, A006116.

Programs

Maple

b:= proc(n) option remember; add(mul(
      (3^(i+k)-1)/(3^i-1), i=1..n-k), k=0..n)
    end:
a:= proc(n) option remember;
      add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 24 2019

Mathematica

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 *)

Formula

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