A045622 Convolution of A000108 (Catalan numbers) with A045543.

1, 25, 362, 3973, 36646, 299530, 2238676, 15613741, 103054094, 650194974, 3950996556, 23257207714, 133217073276, 745218012084, 4083224828328, 21966983072637, 116268166691358, 606474982072982, 3122157367765788

Offset: 1

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A045530 with A000984 (central binomial coefficients); also convolution of A045505 with A000302 (powers of 4).

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Programs

Magma

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

Maple

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

Mathematica

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

PARI

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

Sage

def A045622_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-sqrt(1-4*x))/(2*(1-4*x)^6) ).list()
A045622_list(40) # G. C. Greubel, Jan 13 2020

Formula

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

G.f.: x*c(x)/(1-4*x)^6, where c(x) = g.f. for Catalan numbers.