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A304944 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 16*a(n-2) for n > 1.

%I A304944 #18 Jun 08 2023 15:33:00
%S A304944 0,1,6,34,164,790,3572,16212,71048,312678,1345220,5809980,24692600,
%T A304944 105305980,443684360,1875046120,7848968208,32944100998,137210821092,
%U A304944 572842556332,2376270786840,9878362137364,40842721771544,169192718317336,697620779210096
%N A304944 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 16*a(n-2) for n > 1.
%C A304944 Let a(0) = 0, a(1) = 1 and a(n) = 2*m*a(n-1)/(n-1) + k^2*a(n-2) for n > 1, then the g.f. is x/(2*m) * d/dx ((1 + k*x)/(1 - k*x))^(m/k).
%H A304944 Seiichi Manyama, <a href="/A304944/b304944.txt">Table of n, a(n) for n = 0..1000</a>
%F A304944 a(n) = n*A304941(n)/6.
%F A304944 G.f.: x/(1-4*x)^2 * ((1-4*x)/(1+4*x))^(1/4).
%t A304944 CoefficientList[Series[x/((1-4*x)^(7/4)*(1+4*x)^(1/4)), {x,0,40}], x] (* _G. C. Greubel_, Jun 07 2023 *)
%o A304944 (Magma) [n le 2 select n-1 else 2*(3*Self(n-1) + 8*(n-2)*Self(n-2))/(n-2): n in [1..40]]; // _G. C. Greubel_, Jun 07 2023
%o A304944 (SageMath)
%o A304944 @CachedFunction
%o A304944 def a(n): # b = A304944
%o A304944     if n<2: return n
%o A304944     else: return 2*(3*a(n-1) + 8*(n-1)*a(n-2))//(n-1)
%o A304944 [a(n) for n in range(41)] # _G. C. Greubel_, Jun 07 2023
%Y A304944 Cf. A304933, A304941.
%K A304944 nonn
%O A304944 0,3
%A A304944 _Seiichi Manyama_, May 22 2018