A295611 - OEIS

A295611

a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^k.

%I #12 Nov 25 2017 04:28:25

%S 1,0,0,6,-30,-280,35070,-2508268,-47103462,241470400824,

%T -256752145545390,128291714550379292,2203924344437376054780,

%U -37693423679943326954848176,485163732930867224220253809178,27101025121379607823580070619517816

%N a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^k.

%H Seiichi Manyama, <a href="/A295611/b295611.txt">Table of n, a(n) for n = 0..75</a>

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

%F a(n) = Sum_{k=0..n} (-1)^k*A219206(n,k).

%F Limit n->infinity |a(n)|^(1/n^2) = r^(r^2/(1-2*r)) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243096929661621777... is the real root of the equation (1-r)^(2*r-1) = r^(2*r). - _Vaclav Kotesovec_, Nov 25 2017

%t Table[Sum[(-1)^k Binomial[n, k]^k, {k, 0, n}], {n, 0, 15}]

%t Table[Sum[(-1)^k (n!/(k! (n - k)!))^k, {k, 0, n}], {n, 0, 15}]

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^k); \\ _Michel Marcus_, Nov 25 2017

%Y Cf. A167008, A167010, A219206.

%K sign

%O 0,4

%A _Ilya Gutkovskiy_, Nov 24 2017