A042985 - OEIS

%I #13 Sep 08 2022 08:44:55

%S 1,17,178,1477,10654,69930,428772,2496813,13962982,75582078,398302268,

%T 2052354850,10375356460,51596749300,252953904072,1224672639357,

%U 5863899363510,27801377704310,130648178243660,609082400931158

%N Convolution of A000108 (Catalan numbers) with A038846.

%C Also convolution of A045724 with A000984 (central binomial coefficients); also convolution of A042941 with A000302 (powers of 4).

%H G. C. Greubel, <a href="/A042985/b042985.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = binomial(n+4, 3)*(4^(n+1) - A000984(n+4)/A000984(3))/2, where A000984(n) = binomial(2*n, n).

%F G.f.: (1 - sqrt(1-4*x))/(2*x*(1-4*x)^4).

%F D-finite with recurrence: n*(n+1)*a(n) -2*n*(4*n+13)*a(n-1) +8*(n+3)*(2*n+5)*a(n-2)=0. - _R. J. Mathar_, Jan 28 2020

%t CoefficientList[Series[(1-Sqrt[1-4*x])/(2*x*(1-4*x)^4), {x, 0, 20}], x] (* _G. C. Greubel_, Feb 17 2019 *)

%o (PARI) my(x='x+O('x^20)); Vec((1-sqrt(1-4*x))/(2*x*(1-4*x)^4)) \\ _G. C. Greubel_, Feb 17 2019

%o (Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(2*x*(1-4*x)^4) )); // _G. C. Greubel_, Feb 17 2019

%o (Sage) ((1-sqrt(1-4*x))/(2*x*(1-4*x)^4)).series(x, 20).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 17 2019

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_