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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 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

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Last modified November 26 23:41 EST 2020. Contains 338670 sequences. (Running on oeis4.)