A127850 a(n)=(2^n-1)*2^(n(n-1)/2)/(2^(n-1)).

0, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760, 137170518016, 70300024700928, 72022409665839104, 147537923792657448960, 604389122831019749146624, 4951457925686617442302820352

Offset: 0

Author

Paul Barry, Feb 02 2007

Keywords

• easy
• nonn
• changed

Comments

To base 2, this is given by A127851.

a(n)=(n-1)-st elementary symmetric function of {1,2,4,6,16,...,2^(n-1)}; see Mathematica program. - Clark Kimberling, Dec 29 2011

With offset = 1: the number of simple labeled graphs on n vertices in which vertex 1 or vertex 2 is isolated (or both). - Geoffrey Critzer, Dec 27 2012

HANKEL transform of A001003(n+2) (= [3, 11, 45, ...]) is a(n+2) (= [3, 14, 120, ...]). - Michael Somos, May 19 2013

Examples

			G.f. = x + 3*x^2 + 14*x^3 + 120*x^4 + 1984*x^5 + 64512*x^6 + 4161536*x^7 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..80

Crossrefs

Cf. A001003, A122743, A203011.

Programs

Magma

[2^Binomial( n-1, 2) * (2^n - 1):n in [0..30]]; // Vincenzo Librandi, Aug 31 2014

Mathematica

f[k_] := 2^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
a[n_] := SymmetricPolynomial[n - 1, t[n]]
Table[a[n], {n, 1, 16}]  (* A127850 *)
(* Clark Kimberling, Dec 29 2011 *)
a[ n_] := 2^Binomial[ n - 1, 2] (2^n - 1); (* Michael Somos, Aug 30 2014 *)
Table[2^Binomial[n - 1, 2] (2^n - 1), {n, 0, 30}] (* Vincenzo Librandi, Aug 31 2014 *)
Table[(2^n-1) (2^((n(n-1))/2))/2^(n-1),{n,0,20}] (* Harvey P. Dale, Sep 02 2025 *)

PARI

{a(n) = 2^binomial( n-1, 2) * (2^n - 1)}; /* Michael Somos, Aug 30 2014 */

Formula

a(n) = 2^C(n,2)*(2^n-1)/2^(n-1).

a(-n) = -(4^n) * a(n) for all n in Z. - Michael Somos, Aug 30 2014

0 = +a(n)*(-a(n+2) + a(n+3)) + a(n+1)*(2*a(n+1) - 6*a(n+2) - 4*a(n+3)) + a(n+2)*(+8*a(n+2)) for all n in Z. - Michael Somos, Aug 30 2014

0 = +a(n)*a(n+2)*(-a(n) - 4*a(n+2)) + a(n)*a(n+1)*(+2*a(n+1) + 10*a(n+2)) + a(n+1)^2*(-24*a(n+1) + 8*a(n+2)) for all n in Z. - Michael Somos, Aug 30 2014