A045505 Convolution of A000108 (Catalan numbers) with A040075.

1, 21, 262, 2525, 20754, 152946, 1040556, 6659037, 40599130, 237978598, 1350216660, 7453221490, 40188242420, 212349718980, 1102352779992, 5634083759325, 28400234400810, 141402315307550, 696257439473860

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A045492 with A000984 (central binomial coefficients); also convolution of A042985 with A000302 (powers of 4).

Links

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

Programs

GAP

List([0..20], n-> Binomial(n+5,4)*(2^(2*n+1) - Binomial(2*n+10,n+5)/140)); # G. C. Greubel, Jan 13 2020

Magma

[Binomial(n+5,4)*(2^(2*n+1) - Binomial(2*n+10,n+5)/140): n in [0..20]]; // G. C. Greubel, Jan 13 2020

Maple

seq(coeff(series((1-sqrt(1-4*x))/(2*x*(1-4*x)^5), x, n+1), x, n), n = 0..20); # G. C. Greubel, Jan 13 2020

Mathematica

Table[Binomial[n+5,4]*(2^(2*n+1) -Binomial[2*n+10, n+5]/140), {n,0,20}] (* G. C. Greubel, Jan 13 2020 *)

PARI

vector(21, n, binomial(n+5,4)*(2^(2*n+1) -binomial(2*n+10,n+5)/140)) \\ G. C. Greubel, Jan 13 2020

Sage

[binomial(n+5,4)*(2^(2*n+1) - binomial(2*n+10,n+5)/140) for n in (0..20)] # G. C. Greubel, Jan 13 2020

Formula

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

G.f. c(x)/(1-4*x)^5, where c(x) = g.f. for Catalan numbers.