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

%I A206154 #9 Jan 29 2014 16:55:42
%S A206154 1,2,10,110,2386,125752,14921404,3697835668,2223231412546,
%T A206154 3088517564289836,9040739066816429380,63462297965044771663708,
%U A206154 1064766030857977088480630740,37863276208844960432962611293828,3144384748384240804260912067907833280
%N A206154 a(n) = Sum_{k=0..n} binomial(n,k)^(k+2).
%C A206154 Ignoring initial term a(0), equals the logarithmic derivative of A206153.
%F A206154 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
%e A206154 L.g.f.: L(x) = 2*x + 10*x^2/2 + 110*x^3/3 + 2386*x^4/4 + 125752*x^5/5 +...
%e A206154 where exponentiation yields A206151:
%e A206154 exp(L(x)) = 1 + 2*x + 7*x^2 + 48*x^3 + 693*x^4 + 26632*x^5 + 2542514*x^6 +...
%e A206154 Illustration of initial terms:
%e A206154 a(1) = 1^2 + 1^3 = 2;
%e A206154 a(2) = 1^2 + 2^3 + 1^4 = 10;
%e A206154 a(3) = 1^2 + 3^3 + 3^4 + 1^5 = 110;
%e A206154 a(4) = 1^2 + 4^3 + 6^4 + 4^5 + 1^6 = 2386;
%e A206154 a(5) = 1^2 + 5^3 + 10^4 + 10^5 + 5^6 + 1^7 = 125752; ...
%t A206154 Table[Sum[Binomial[n,k]^(k+2),{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jan 16 2014 *)
%o A206154 (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(k+2))}
%o A206154 for(n=0,16,print1(a(n),", "))
%Y A206154 Cf. A206153 (exp), A184731, A206156, A206158, A206152, A167008, A220359.
%K A206154 nonn
%O A206154 0,2
%A A206154 _Paul D. Hanna_, Feb 04 2012