A076657 a(n) = (1/24) * binomial(2n,n)*(16^n-binomial(2n,n)^2). Right side of identity involving series A005148.

0, 1, 55, 3080, 176855, 10343256, 613052440, 36701926976, 2214353424855, 134425330290680, 8201448540559560, 502460159228920256, 30890758976011469080, 1904794982716556862400, 117756015163729064222400, 7296082202981986626900480, 452950299939910627966962135

Offset: 0

Author

Ralf Stephan, Oct 24 2002

Keywords

• nonn

Comments

The members of the sequence have exceptionally many small prime factors.

Examples

			G.f. = x + 55*x^2 + 3080*x^3 + 176855*x^4 + 10343256*x^5 + 613052440*x^6 + ...
		

References

• D. Shanks, Solved and unsolved problems in number theory, Chelsea NY, 1985, pp. 256-257 (F. Beukers, Letter to D. Shanks, Mar. 13, 1984).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A005148.

Programs

Magma

[(Binomial(2*n, n)*(16^n-Binomial(2*n, n)^2))/24 : n in [0..20]]; // Vincenzo Librandi, May 17 2013

Mathematica

a[n_] := (Binomial[2n, n]*(16^n-Binomial[2n, n]^2))/24
Table[(Binomial[2 n, n] (16^n - Binomial[2 n, n]^2)) / 24, {n, 0, 20}] (* Vincenzo Librandi, May 17 2013 *)

PARI

{a(n) = if( n<0, 0, (binomial(2*n, n) * (16^n - binomial(2*n, n)^2)) / 24)};

Formula

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).