%I #7 Mar 03 2014 14:06:34

%S 1,2,10,272,24226,12053252,40086916024,429254371605824,

%T 23527609330364490754,10714627376371224032350052,

%U 16964729291782419425708732425300,109783535843179466164398767001178968704,6782057095273243388704415924996348722446049600

%N a(n) = Sum_{k=0..n} binomial(n,k)^(2*k+1).

%C Ignoring initial term a(0), equals the logarithmic derivative of A206157.

%F Limit n->infinity a(n)^(1/n^2) = r^(2*r^2/(1-2*r)) = 2.3520150420944489879258119..., where r = 0.70350607643066243... (see A220359) is the root of the equation (1-r)^(2*r-1) = r^(2*r). - _Vaclav Kotesovec_, Mar 03 2014

%e L.g.f.: L(x) = 2*x + 10*x^2/2 + 272*x^3/3 + 24226*x^4/4 + 12053252*x^5/5 +...

%e where exponentiation yields A206157:

%e exp(L(x)) = 1 + 2*x + 7*x^2 + 102*x^3 + 6261*x^4 + 2423430*x^5 + 6686021554*x^6 +...

%e Illustration of initial terms:

%e a(1) = 1^1 + 1^3 = 2;

%e a(2) = 1^1 + 2^3 + 1^5 = 10;

%e a(3) = 1^1 + 3^3 + 3^5 + 1^7 = 272;

%e a(4) = 1^1 + 4^3 + 6^5 + 4^7 + 1^9 = 24226;

%e a(5) = 1^1 + 5^3 + 10^5 + 10^7 + 5^9 + 1^11 = 12053252; ...

%t Table[Sum[Binomial[n,k]^(2*k+1), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(2*k+1))}

%o for(n=0,16,print1(a(n),", "))

%Y Cf. A206157 (exp), A184731, A206154, A206156, A206152, A220359.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2012