A045492 - OEIS

%I #8 Sep 08 2022 08:44:56

%S 1,19,218,1955,15086,105102,679764,4154403,24281510,136887322,

%T 749032492,3997228430,20880823820,107088473660,540472210728,

%U 2689562860323,13217998697430,64240718824930,309108505173820

%N Convolution of A000108 (Catalan numbers) with A020920.

%C Also convolution of A042985 with A000984 (central binomial coefficients); also convolution of A045724 with A000302 (powers of 4).

%H G. C. Greubel, <a href="/A045492/b045492.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%p 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

%t 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 *)

%o (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

%o (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

%o (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

%o (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

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_