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

0,3

Comments

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

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

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).

Example

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

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 *)

Prog

(PARI) {a(n) = if( n<0, 0, (binomial(2*n, n) * (16^n - binomial(2*n, n)^2)) / 24)};
(Magma) [(Binomial(2*n, n)*(16^n-Binomial(2*n, n)^2))/24 : n in [0..20]]; // Vincenzo Librandi, May 17 2013

Crossrefs

Cf. A005148.

Sequence in context: A119166 A027548 A231640 * A212736 A221767 A095659

Adjacent sequences: A076654 A076655 A076656 * A076658 A076659 A076660

Keyword

nonn

Author

Ralf Stephan, Oct 24 2002

Status

approved