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A046714 Convolution of A000108 (Catalan) with A000351 (powers of 5).

%I A046714 #13 Jul 29 2024 06:22:49
%S A046714 1,6,32,165,839,4237,21317,107014,536500,2687362,13453606,67326816,
%T A046714 336842092,1684953360,8427441240,42146901045,210769862895,
%U A046714 1053978959265,5270372435025,26353629438315,131774711311995
%N A046714 Convolution of A000108 (Catalan) with A000351 (powers of 5).
%H A046714 G. C. Greubel, <a href="/A046714/b046714.txt">Table of n, a(n) for n = 0..1000</a>
%F A046714 a(n) = Sum_{k=0..n} A000108(k)*5^(n-k).
%F A046714 a(n) = 5*a(n-1) + C(n), a(0) = 1.
%F A046714 G.f.: c(x)/(1-5*x), where c(x) = g.f. for Catalan numbers A000108.
%F A046714 Homogeneous recursion: a(n) = (3*(3*n+1)/(n+1))*a(n-1) - (10*(2*n-1)/(n+1))*a(n-2), a(-1) := 0, a(0)=1, n >= 1.
%F A046714 Hypergeometric 2F1 form: 2*a(n) = 5^(n+1) - binomial(2*(n+1), n+1) * hypergeom([ -n-1, 1 ], [ 1/2 ], -1/4).
%F A046714 a(n) ~ (5-sqrt(5))/2 * 5^n. - _Vaclav Kotesovec_, Jul 07 2016
%t A046714 CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-5*x)), {x,0,40}], x] (* _G. C. Greubel_, Jul 28 2024 *)
%o A046714 (Magma)
%o A046714 [n le 1 select 1 else 5*Self(n-1) + Catalan(n-1): n in [1..40]]; // _G. C. Greubel_, Jul 28 2024
%o A046714 (SageMath)
%o A046714 @CachedFunction
%o A046714 def A046714(n): return 1 if n==1 else 5*A046714(n-1) + catalan_number(n-1)
%o A046714 [A046714(n) for n in range(1,41)] # _G. C. Greubel_, Jul 28 2024
%Y A046714 Cf. A000108, A000351.
%K A046714 easy,nonn
%O A046714 0,2
%A A046714 _Wolfdieter Lang_