A045530 Convolution of A000108 (Catalan numbers) with A020922.

1, 23, 310, 3195, 27866, 216566, 1546028, 10338515, 65635570, 399429602, 2346750900, 13384232030, 74417751940, 404759481420, 2159510136408, 11327603405955, 58528412321250, 298354368109930, 1502525977613540

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A045505 with A000984 (central binomial coefficients); also convolution of A045492 with A000302 (powers of 4).

Links

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

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (Sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) )); // G. C. Greubel, Jan 13 2020

Maple

seq(coeff(series((sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6), x, n+1), x, n), n = 0..40); # G. C. Greubel, Jan 13 2020

Mathematica

CoefficientList[Series[(Sqrt[1-4*x] +4*x-1)/(2*x*(1-4*x)^6), {x,0,40}], x] (* G. C. Greubel, Jan 13 2020 *)

PARI

my(x='x+O('x^40)); Vec( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ) \\ G. C. Greubel, Jan 13 2020

Sage

def A045530_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (sqrt(1-4*x) +4*x-1)/(2*x*(1-4*x)^6) ).list()
A045530_list(40) # G. C. Greubel, Jan 13 2020

Formula

a(n) = binomial(n+6, 5)*(A000984(n+6)/A000984(5) - 5*4^(n+1)/(n+6))/2, A000984(n) = binomial(2*n, n).

G.f. c(x)/(1-4*x)^(11/2), where c(x) = g.f. for Catalan numbers.