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A305032 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 4*a(n-2) for n > 1.

%I A305032 #15 Jun 07 2023 20:06:05
%S A305032 0,1,6,22,68,190,500,1260,3080,7350,17220,39732,90552,204204,456456,
%T A305032 1012440,2230800,4886310,10647780,23094500,49884120,107343236,
%U A305032 230205976,492156392,1049212528,2230928700,4732273000,10015777800,21154820400,44596287000,93846099600
%N A305032 a(0) = 0, a(1) = 1 and a(n) = 6*a(n-1)/(n-1) + 4*a(n-2) for n > 1.
%C A305032 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 A305032 Seiichi Manyama, <a href="/A305032/b305032.txt">Table of n, a(n) for n = 0..3000</a>
%F A305032 a(n) = n*A305031(n)/6.
%F A305032 G.f.: x*sqrt(1-4*x^2)/(1-2*x)^3.
%t A305032 CoefficientList[Series[x*Sqrt[1-4*x^2]/(1-2*x)^3, {x,0,40}], x] (* _G. C. Greubel_, Jun 07 2023 *)
%o A305032 (Magma) [n le 2 select n-1 else 2*(3*Self(n-1) + 2*(n-2)*Self(n-2))/(n-2): n in [1..40]]; // _G. C. Greubel_, Jun 07 2023
%o A305032 (SageMath)
%o A305032 @CachedFunction
%o A305032 def a(n): # b = A305032
%o A305032     if n<2: return n
%o A305032     else: return 2*(3*a(n-1) + 2*(n-1)*a(n-2))//(n-1)
%o A305032 [a(n) for n in range(41)] # _G. C. Greubel_, Jun 07 2023
%Y A305032 Cf. A100071, A304944, A305031.
%K A305032 nonn
%O A305032 0,3
%A A305032 _Seiichi Manyama_, May 24 2018