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%I #17 Sep 08 2022 08:45:29
%S 1,7,56,455,3710,30282,247254,2019087,16488710,134656130,1099686056,
%T 8980749862,73342721956,598965319960,4891549246290,39947649057855,
%U 326239122661830,2664286127154330,21758336553841440,177693081299126610
%N Expansion of g.f.: 1/(1 - 7*x*c(x)), where c(x) is the g.f. for A000108.
%C The Hankel transform of this sequence is 7^n = [1, 7, 49, 343, 2401, ...] . The Hankel transform of the aerated sequence with g.f. 1/(1 - 7*x^2*c(x^2)) is also 7^n.
%C Numbers have the same parity as the Catalan numbers, that is, a(n) is even except for n of the form 2^m - 1. Follows from c(x) = 1/(1 - x*c(x)) == 1/(1 - 7*x*c(x)) (mod 2). - _Peter Bala_, Jul 24 2016
%H G. C. Greubel, <a href="/A126694/b126694.txt">Table of n, a(n) for n = 0..1000</a>
%F a(0) = 1, a(n) = (49*a(n-1) - 7*A000108(n-1))/6 for n >= 1.
%F a(n) = Sum_{k = 0..n} A106566(n,k)*7^k.
%F a(n) = Sum_{k = 0..n} A039599(n,k)*6^k.
%F a(n) ~ 5 * 7^(2*n) / 6^(n+1). - _Vaclav Kotesovec_, Nov 29 2021
%t CoefficientList[Series[2/(-5+7*Sqrt[1-4*x]), {x, 0, 30}], x] (* _G. C. Greubel_, May 05 2019 *)
%o (PARI) my(x='x+O('x^30)); Vec(2/(7*sqrt(1-4*x) -5)) \\ _G. C. Greubel_, May 05 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(7*Sqrt(1-4*x) -5) )); // _G. C. Greubel_, May 05 2019
%o (Sage) (2/(7*sqrt(1-4*x) -5)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 05 2019
%Y Cf. A000108, A000984, A007854, A076035, A076036, A127628, A115970.
%K nonn,easy
%O 0,2
%A _Philippe Deléham_, Feb 14 2007
%E a(16) corrected by _G. C. Greubel_, May 05 2019
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