A042985 Convolution of A000108 (Catalan numbers) with A038846.

1, 17, 178, 1477, 10654, 69930, 428772, 2496813, 13962982, 75582078, 398302268, 2052354850, 10375356460, 51596749300, 252953904072, 1224672639357, 5863899363510, 27801377704310, 130648178243660, 609082400931158

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-Sqrt(1-4*x))/(2*x*(1-4*x)^4) )); // G. C. Greubel, Feb 17 2019

Mathematica

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

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

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

Formula

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

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

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