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%I #24 Sep 08 2022 08:45:07
%S 0,1,55,3080,176855,10343256,613052440,36701926976,2214353424855,
%T 134425330290680,8201448540559560,502460159228920256,
%U 30890758976011469080,1904794982716556862400,117756015163729064222400,7296082202981986626900480,452950299939910627966962135
%N a(n) = (1/24) * binomial(2n,n)*(16^n-binomial(2n,n)^2). Right side of identity involving series A005148.
%C The members of the sequence have exceptionally many small prime factors.
%D D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, pp. 256-257 (F. Beukers, Letter to D. Shanks, Mar. 13, 1984).
%H Vincenzo Librandi, <a href="/A076657/b076657.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = (1/24) * binomial(2n, n)*(16^n-binomial(2n, n)^2) = Sum_{i=1..n} binomial(2n-2i, n-i)^3 * A005148(i) (Shanks and Beukers).
%e G.f. = x + 55*x^2 + 3080*x^3 + 176855*x^4 + 10343256*x^5 + 613052440*x^6 + ...
%t a[n_] := (Binomial[2n, n]*(16^n-Binomial[2n, n]^2))/24
%t Table[(Binomial[2 n, n] (16^n - Binomial[2 n, n]^2)) / 24, {n, 0, 20}] (* _Vincenzo Librandi_, May 17 2013 *)
%o (PARI) {a(n) = if( n<0, 0, (binomial(2*n, n) * (16^n - binomial(2*n, n)^2)) / 24)};
%o (Magma) [(Binomial(2*n, n)*(16^n-Binomial(2*n, n)^2))/24 : n in [0..20]]; // _Vincenzo Librandi_, May 17 2013
%Y Cf. A005148.
%K nonn
%O 0,3
%A _Ralf Stephan_, Oct 24 2002
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