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A127850 a(n)=(2^n-1)*2^(n(n-1)/2)/(2^(n-1)). 4
0, 1, 3, 14, 120, 1984, 64512, 4161536, 534773760, 137170518016, 70300024700928, 72022409665839104, 147537923792657448960, 604389122831019749146624, 4951457925686617442302820352 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

LINKS

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

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

EXAMPLE

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

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 *)

PROG

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

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

CROSSREFS

Cf. A001003, A122743, A203011.

Sequence in context: A161936 A304983 A260887 * A324147 A186772 A061029

Adjacent sequences:  A127847 A127848 A127849 * A127851 A127852 A127853

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 02 2007

STATUS

approved

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Last modified October 19 03:34 EDT 2019. Contains 328211 sequences. (Running on oeis4.)