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A046814 Row sums of triangle A046527.

%I A046814 #16 Jul 29 2024 06:23:08
%S A046814 1,2,8,37,179,881,4369,21746,108444,541362,2704158,13512392,67534828,
%T A046814 337584992,1687627800,8437136085,42182258715,210899507685,
%U A046814 1054456597965,5272139698215,26360193558735,131799177579015
%N A046814 Row sums of triangle A046527.
%H A046814 G. C. Greubel, <a href="/A046814/b046814.txt">Table of n, a(n) for n = 0..1000</a>
%F A046814 G.f.: c(x) * (1-4*x) / (1-5*x), where c(x) = g.f. for Catalan A000108.
%F A046814 a(n) = C(n) + A046714(n-1) with A046714(-1) = 0 and C(n) = A000108(n) are the Catalan numbers.
%F A046814 a(n) = C(n) + (5^n - A046748(n))/2.
%F A046814 a(n) = 5*a(n-1) - 3*C(n)/(2*n-1), a(0)=1.
%F A046814 D-finite with recurrence a(n) = (9*n-1)*a(n-1)/(n+1) - 10*(2*n-3)*a(n-2)/(n+1), n >= 2, a(0)=1, a(1)=2.
%t A046814 CoefficientList[Series[(1-4*x)*(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x,0,40}], x] (* _G. C. Greubel_, Jul 28 2024 *)
%o A046814 (Magma)
%o A046814 [n le 1 select 1 else 5*Self(n-1) - 3*Catalan(n-1)/(2*n-3): n in [1..40]]; // _G. C. Greubel_, Jul 28 2024
%o A046814 (SageMath)
%o A046814 @CachedFunction
%o A046814 def A046814(n): return 1 if n==0 else 5*A046814(n-1) - 3*catalan_number(n)/(2*n-1)
%o A046814 [A046814(n) for n in range(41)] # _G. C. Greubel_, Jul 28 2024
%Y A046814 Cf. A000108, A046527, A046714, A046748.
%K A046814 easy,nonn
%O A046814 0,2
%A A046814 _Wolfdieter Lang_
%E A046814 Offset corrected by _Sean A. Irvine_, Apr 25 2021