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A076657 a(n) = (1/24) * binomial(2n,n)*(16^n-binomial(2n,n)^2). Right side of identity involving series A005148. 2
0, 1, 55, 3080, 176855, 10343256, 613052440, 36701926976, 2214353424855, 134425330290680, 8201448540559560, 502460159228920256, 30890758976011469080, 1904794982716556862400, 117756015163729064222400, 7296082202981986626900480, 452950299939910627966962135 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Ralf Stephan, Oct 24 2002
STATUS
approved

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Last modified July 20 09:02 EDT 2024. Contains 374445 sequences. (Running on oeis4.)