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%I #14 Sep 24 2019 18:54:38
%S 1,1,3,15,129,1833,43347,1705623,112931553,12639552945,2413134909507,
%T 788041911546303,442817851480763169,428369525248261655193,
%U 716160018275094098267859,2067365673240491189928496263,10333740296321620864171488891201
%N Inverse binomial transform of the Galois numbers G_(n)^{(3)} (A006117).
%C Analog of the inverse binomial transform of G_(n)^{(q)} with q=2, A135922.
%H Alois P. Heinz, <a href="/A228365/b228365.txt">Table of n, a(n) for n = 0..91</a>
%F 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 b:= proc(n) option remember; add(mul(
%p (3^(i+k)-1)/(3^i-1), i=1..n-k), k=0..n)
%p end:
%p a:= proc(n) option remember;
%p add(b(n-j)*binomial(n, j)*(-1)^j, j=0..n)
%p end:
%p seq(a(n), n=0..19); # _Alois P. Heinz_, Sep 24 2019
%t 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 Cf. A135922, A006117, A006116.
%K nonn
%O 0,3
%A _R. J. Mathar_, Aug 21 2013
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