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%I #9 Mar 03 2014 13:51:39
%S 1,2,6,92,5410,1400652,2687407464,18947436116184,536104663173431874,
%T 130559883231879141946580,136031455187223511721647272376,
%U 483565526783420050082035900177878504,14487924180895151383693101563813954330590756
%N a(n) = Sum_{k=0..n} binomial(n,k)^(2*k).
%C Ignoring initial term a(0), equals the logarithmic derivative of A206155.
%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 + 6*x^2/2 + 92*x^3/3 + 5410*x^4/4 + 1400652*x^5/5 +...
%e where exponentiation yields A206155:
%e exp(L(x)) = 1 + 2*x + 5*x^2 + 38*x^3 + 1425*x^4 + 283002*x^5 + 448468978*x^6 +...
%e Illustration of initial terms:
%e a(1) = 1^0 + 1^2 = 2;
%e a(2) = 1^0 + 2^2 + 1^4 = 6;
%e a(3) = 1^0 + 3^2 + 3^4 + 1^6 = 92;
%e a(4) = 1^0 + 4^2 + 6^4 + 4^6 + 1^8 = 5410;
%e a(5) = 1^0 + 5^2 + 10^4 + 10^6 + 5^8 + 1^10 = 1400652; ...
%t Table[Sum[Binomial[n,k]^(2*k), {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)^(2*k))}
%o for(n=0,16,print1(a(n),", "))
%Y Cf. A206155 (exp), A184731, A206154, A206158, A206152, A220359.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 04 2012
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