A095338 - OEIS

%I #32 Sep 08 2022 08:45:13

%S 0,2,12,96,1280,30720,1376256,117440512,19327352832,6184752906240,

%T 3870280929771520,4755801206503243776,11510768301994760208384,

%U 55006124792465627449131008,519934816499859715457632174080,9735556609752801803494680617287680,361550014853497117429835520396253724672

%N Total number of leaves in the labeled graphs of order n.

%C A leaf is defined as a vertex of degree (or valence) 1. - _Michael Somos_, Mar 13 2014

%H Vincenzo Librandi, <a href="/A095338/b095338.txt">Table of n, a(n) for n = 1..80</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TreeLeaf.html">Tree Leaf</a>

%F Conjecture: a(n) = n*(n-1)*2^binomial(n-1,2). - _Vladeta Jovovic_, Jan 26 2006

%F a(n) = n*(n-1)*2^binomial(n-1,2) is correct, since counting the total number of leaves in the labeled graphs of order n is equivalent to counting all labeled rooted graphs of order n where the root is a leaf. - _Bertran Steinsky_, Mar 04 2014

%F a(n) = 2^(n-1) * A182166(n) for n>=2. - _Joerg Arndt_, Mar 12 2014

%e G.f. = 2*x^2 + 12*x^3 + 96*x^4 + 1280*x^5 + 30720*x^6 + 1376256*x^7 + ...

%p A095338:=n->n*(n-1)*2^binomial(n-1,2): seq(A095338(n), n=1..20); # _Wesley Ivan Hurt_, Oct 17 2014

%t Table[n (n - 1) 2^(Binomial[n-1, 2]), {n, 20}] (* _Vincenzo Librandi_, Mar 14 2014 *)

%o (PARI) a(n) = n*(n-1)*2^binomial(n-1,2); \\ _Joerg Arndt_, Mar 12 2014

%o (Magma) [n*(n-1)*2^Binomial(n-1, 2): n in [1..20]]; // _Vincenzo Librandi_, Mar 14 2014

%Y Cf. A182166.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_, Jun 02 2004