A295611 a(n) = Sum_{k=0..n} (-1)^k*binomial(n,k)^k.

1, 0, 0, 6, -30, -280, 35070, -2508268, -47103462, 241470400824, -256752145545390, 128291714550379292, 2203924344437376054780, -37693423679943326954848176, 485163732930867224220253809178, 27101025121379607823580070619517816

Offset: 0

Author

Ilya Gutkovskiy, Nov 24 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..75
• Eric Weisstein's World of Mathematics, Binomial Sums

Crossrefs

Cf. A167008, A167010, A219206.

Programs

Mathematica

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

PARI

a(n) = sum(k=0, n, (-1)^k*binomial(n,k)^k); \\ Michel Marcus, Nov 25 2017

Formula

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

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