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A156195 a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1.

%I A156195 #15 Nov 09 2022 19:17:43
%S A156195 1,5,30,175,1050,6250,37500,224375,1346250,8068750,48412500,290343750,
%T A156195 1742062500,10450312500,62701875000,376177734375,2257066406250,
%U A156195 13541839843750,81251039062500,487496738281250,2924980429687500,17549718554687500,105298311328125000
%N A156195 a(2n+2) = 6*a(2n+1), a(2n+1) = 6*a(2n) - 5^n*A000108(n), a(0)=1.
%C A156195 Hankel transform is 5^C(n+1,2). - _Philippe Deléham_, Feb 05 2009
%H A156195 G. C. Greubel, <a href="/A156195/b156195.txt">Table of n, a(n) for n = 0..1000</a>
%H A156195 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Barry2/barry73.html">A Note on a One-Parameter Family of Catalan-Like Numbers</a>, JIS 12 (2009) 09.5.4.
%F A156195 a(n) = Sum_{k=0..n} A120730(n,k)*5^k.
%F A156195 G.f.: (sqrt(1-20*x^2) +10*x -1)/(10*x*(1-6*x)). - _Philippe Deléham_, Feb 05 2009
%F A156195 (n+1)*a(n) = 6*(n+1)*a(n-1) + 20*(n-2)*a(n-2) - 120*(n-2)*a(n-3). - _R. J. Mathar_, Jul 21 2016
%p A156195 A156195 := proc(n)
%p A156195     option remember;
%p A156195     local nh;
%p A156195     if n= 0 then
%p A156195         1;
%p A156195     elif  type(n,'even') then
%p A156195         6*procname(n-1);
%p A156195     else
%p A156195         nh := floor(n/2) ;
%p A156195         6*procname(n-1)-5^nh*A000108(nh) ;
%p A156195     end if;
%p A156195 end proc: # _R. J. Mathar_, Jul 21 2016
%t A156195 CoefficientList[Series[(Sqrt[1-20x^2]+10x-1)/(10x(1-6x)),{x,0,30}],x] (* _Harvey P. Dale_, Oct 21 2016 *)
%o A156195 (Magma) [n le 3 select Factorial(n+3)/24 else (6*n*Self(n-1) + 20*(n-3)*Self(n-2) - 120*(n-3)*Self(n-3))/n: n in [1..30]]; // _G. C. Greubel_, Nov 09 2022
%o A156195 (SageMath)
%o A156195 def a(n): # a = A156195
%o A156195     if (n==0): return 1
%o A156195     elif (n%2==1): return 6*a(n-1) - 5^((n-1)/2)*catalan_number((n-1)/2)
%o A156195     else: return 6*a(n-1)
%o A156195 [a(n) for n in (0..30)] # _G. C. Greubel_, Nov 09 2022
%Y A156195 Cf. A000108, A001405, A120730, A151162, A151254, A151281, A156058.
%K A156195 nonn
%O A156195 0,2
%A A156195 _Philippe Deléham_, Feb 05 2009
%E A156195 Corrected and extended by _Harvey P. Dale_, Oct 21 2016