A206154 a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).

1, 2, 10, 110, 2386, 125752, 14921404, 3697835668, 2223231412546, 3088517564289836, 9040739066816429380, 63462297965044771663708, 1064766030857977088480630740, 37863276208844960432962611293828, 3144384748384240804260912067907833280

Offset: 0

Author

Paul D. Hanna, Feb 04 2012

Keywords

• nonn

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A206153.

Examples

			L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
where exponentiation yields A206151:
exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
Illustration of initial terms:
a(1) = 1^2 + 1^3 = 2;
a(2) = 1^2 + 2^3 + 1^4 = 10;
a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...
		

Crossrefs

Cf. A206153 (exp), A184731, A206156, A206158, A206152, A167008, A220359.

Programs

Mathematica

Table[Sum[Binomial[n,k]^(k+2),{k,0,n}],{n,0,20}] (* Harvey P. Dale, Jan 16 2014 *)

PARI

{a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
for(n=0,16,print1(a(n),", "))

Formula

Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.53362806511..., where r = 0.70350607643... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - Vaclav Kotesovec, Jan 29 2014