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%I #8 Sep 08 2022 08:44:56
%S 1,21,262,2525,20754,152946,1040556,6659037,40599130,237978598,
%T 1350216660,7453221490,40188242420,212349718980,1102352779992,
%U 5634083759325,28400234400810,141402315307550,696257439473860
%N Convolution of A000108 (Catalan numbers) with A040075.
%C Also convolution of A045492 with A000984 (central binomial coefficients); also convolution of A042985 with A000302 (powers of 4).
%H G. C. Greubel, <a href="/A045505/b045505.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = binomial(n+5, 4)*(4^(n+1) - A000984(n+5)/A000984(4))/2, A000984(n) = binomial(2*n, n).
%F G.f. c(x)/(1-4*x)^5, where c(x) = g.f. for Catalan numbers.
%p 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
%t 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 *)
%o (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
%o (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
%o (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
%o (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
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_
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