A045492 Convolution of A000108 (Catalan numbers) with A020920.

1, 19, 218, 1955, 15086, 105102, 679764, 4154403, 24281510, 136887322, 749032492, 3997228430, 20880823820, 107088473660, 540472210728, 2689562860323, 13217998697430, 64240718824930, 309108505173820

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Comments

Also convolution of A042985 with A000984 (central binomial coefficients); also convolution of A045724 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)*(Binomial(2*n+10, n+5)/140 - 2^(2*n+3)/(n+5))); # G. C. Greubel, Jan 13 2020

Magma

[Binomial(n+5, 4)*(Binomial(2*n+10, n+5)/140 - 2^(2*n+3)/(n+5)): n in [0..20]]; // G. C. Greubel, Jan 13 2020

Maple

seq(coeff(series((sqrt(1-4*x) +4*x-1)/(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]*(Binomial[2*n+10, n+5]/140 - 2^(2*n+3)/(n+5)), {n,0,20}] (* G. C. Greubel, Jan 13 2020 *)

PARI

vector(20, n, binomial(n+4, 4)*(binomial(2*n+8, n+4)/140 - 2^(2*n+1)/(n+4)) ) \\ G. C. Greubel, Jan 13 2020

Sage

[binomial(n+5, 4)*(binomial(2*n+10, n+5)/140 - 2^(2*n+3)/(n+5)) for n in (0..20)] # G. C. Greubel, Jan 13 2020

Formula

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

G.f.: c(x)/(1-4*x)^(9/2) = (2-c(x))/(1-4*x)^5, where c(x) = g.f. for Catalan numbers.