A046885 - OEIS

%I #8 Jul 29 2024 06:23:14

%S 1,4,18,85,411,2013,9933,49236,244750,1218888,6077644,30329434,

%T 151439158,756452890,3779590010,18888255205,94405918355,471899946985,

%U 2359022096225,11793343217935,58960151969255,294776293579255

%N Row sums of triangle A046658.

%H G. C. Greubel, <a href="/A046885/b046885.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = 2*5^(n-1) - A046714(n-1) = (A046748(n) - 5^(n-1))/2.

%F G.f.: x*(2 - c(x))/(1-5*x), where c(x) is the g.f. of A000108 (Catalan numbers).

%F Inhomogeneous recursion: a(n) = 5*a(n-1) - C(n-1), n >= 2, a(1)=1; C(n) = A000108(n) (Catalan).

%F Homogeneous recursion: a(n) = (3*(3*n-2)/n)*a(n-1) - (10*(2*n-3)/n)*a(n-2), n >= 3, a(1)=1, a(2)=4.

%t Rest@CoefficientList[Series[Sqrt[1-4*x]*(1-Sqrt[1-4*x])/(2*(1-5*x)), {x,0,40}], x] (* _G. C. Greubel_, Jul 28 2024 *)

%o (Magma)

%o [n le 1 select 1 else 5*Self(n-1) - Catalan(n-1): n in [1..40]]; // _G. C. Greubel_, Jul 28 2024

%o (SageMath)

%o @CachedFunction

%o def A046885(n): return 1 if n==1 else 5*A046885(n-1) - catalan_number(n-1)

%o [A046885(n) for n in range(1,41)] # _G. C. Greubel_, Jul 28 2024

%Y Cf. A000108, A046658, A046714, A046748.

%K easy,nonn

%O 1,2

%A _Wolfdieter Lang_