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

%I A127850 #30 Sep 02 2025 17:11:28
%S A127850 0,1,3,14,120,1984,64512,4161536,534773760,137170518016,
%T A127850 70300024700928,72022409665839104,147537923792657448960,
%U A127850 604389122831019749146624,4951457925686617442302820352
%N A127850 a(n)=(2^n-1)*2^(n(n-1)/2)/(2^(n-1)).
%C A127850 To base 2, this is given by A127851.
%C A127850 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
%C A127850 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
%C A127850 HANKEL transform of A001003(n+2) (= [3, 11, 45, ...]) is a(n+2) (= [3, 14, 120, ...]). - _Michael Somos_, May 19 2013
%H A127850 Vincenzo Librandi, <a href="/A127850/b127850.txt">Table of n, a(n) for n = 0..80</a>
%F A127850 a(n) = 2^C(n,2)*(2^n-1)/2^(n-1).
%F A127850 a(-n) = -(4^n) * a(n) for all n in Z. - _Michael Somos_, Aug 30 2014
%F A127850 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
%F A127850 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
%e A127850 G.f. = x + 3*x^2 + 14*x^3 + 120*x^4 + 1984*x^5 + 64512*x^6 + 4161536*x^7 + ...
%t A127850 f[k_] := 2^(k - 1); t[n_] := Table[f[k], {k, 1, n}]
%t A127850 a[n_] := SymmetricPolynomial[n - 1, t[n]]
%t A127850 Table[a[n], {n, 1, 16}]  (* A127850 *)
%t A127850 (* _Clark Kimberling_, Dec 29 2011 *)
%t A127850 a[ n_] := 2^Binomial[ n - 1, 2] (2^n - 1); (* _Michael Somos_, Aug 30 2014 *)
%t A127850 Table[2^Binomial[n - 1, 2] (2^n - 1), {n, 0, 30}] (* _Vincenzo Librandi_, Aug 31 2014 *)
%t A127850 Table[(2^n-1) (2^((n(n-1))/2))/2^(n-1),{n,0,20}] (* _Harvey P. Dale_, Sep 02 2025 *)
%o A127850 (PARI) {a(n) = 2^binomial( n-1, 2) * (2^n - 1)}; /* _Michael Somos_, Aug 30 2014 */
%o A127850 (Magma) [2^Binomial( n-1, 2) * (2^n - 1):n in [0..30]]; // _Vincenzo Librandi_, Aug 31 2014
%Y A127850 Cf. A001003, A122743, A203011.
%K A127850 easy,nonn,changed
%O A127850 0,3
%A A127850 _Paul Barry_, Feb 02 2007