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A209358 G.f.: (1-4*x)^(-1/4) * (1-8*x)^(-1/8).

%I A209358 #20 Sep 08 2022 08:46:01
%S A209358 1,2,8,40,228,1416,9312,63648,446760,3195728,23179840,169929280,
%T A209358 1256234720,9350462400,69993150720,526455847680,3976132184160,
%U A209358 30138433333440,229168000121600,1747455531216640,13358199405416320,102345801274115840,785740341422453760
%N A209358 G.f.: (1-4*x)^(-1/4) * (1-8*x)^(-1/8).
%H A209358 Iain Fox, <a href="/A209358/b209358.txt">Table of n, a(n) for n = 0..1111</a> (first 201 terms from Vincenzo Librandi)
%F A209358 Equals the self-convolution square root of A209200, which equals the convolution of sequences A000984 and A004981.
%F A209358 Recurrence: n*a(n) = 2*(6*n-5)*a(n-1) - 4*(8*n-13)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012
%F A209358 a(n) ~ Gamma(7/8)*2^(1/4)*8^n*sin(Pi/8)/(n^(7/8)*Pi). - _Vaclav Kotesovec_, Oct 20 2012
%e A209358 G.f.: A(x) = 1 + 2*x + 8*x^2 + 40*x^3 + 228*x^4 + 1416*x^5 + 9312*x^6 +...
%e A209358 such that the square of the g.f. A(x) equals the g.f. of A209200:
%e A209358 A(x)^2 = 1 + 4*x + 20*x^2 + 112*x^3 + 680*x^4 + 4384*x^5 + 29536*x^6 +...
%e A209358 Sequence A209200 equals the convolution of the sequences:
%e A209358 A000984 = [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...],
%e A209358 A004981 = [1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, ...].
%t A209358 CoefficientList[Series[(1-4*x)^(-1/4)*(1-8*x)^(-1/8), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o A209358 (PARI) {a(n)=polcoeff((1-4*x +x*O(x^n))^(-1/4)*(1-8*x +x*O(x^n))^(-1/8), n)}
%o A209358 for(n=0,30,print1(a(n),", "))
%o A209358 (Magma) I:=[2,8]; [1] cat [n le 2 select I[n] else (2*(6*n-5)*Self(n-1) - 4*(8*n-13)*Self(n-2))/n: n in [1..30]]; // _G. C. Greubel_, Jan 03 2018
%Y A209358 Cf. A209200, A000984, A004981.
%K A209358 nonn
%O A209358 0,2
%A A209358 _Paul D. Hanna_, Mar 06 2012