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%I #18 Jan 19 2019 06:57:38
%S 1,2,4,20,296,10067,927100,219541877,110728186648,137502766579907,
%T 448577320868198789,3169529341990169816462,51243646781214826181569316,
%U 2201837465728010770618930322223,215520476721579201896200887266792583,45634827026091489574547858030506357191920
%N a(n) = Sum_{k=0..n} binomial(n*k, k^2).
%C Ignoring initial term, equals the logarithmic derivative of A228809.
%C Equals row sums of triangle A228832.
%H Seiichi Manyama, <a href="/A228808/b228808.txt">Table of n, a(n) for n = 0..73</a>
%F Limit n->infinity a(n)^(1/n^2) = (1-r)^(-r/2) = 1.533628065110458582053143..., where r = A220359 = 0.70350607643066243... is the root of the equation (1-r)^(2*r-1) = r^(2*r). - _Vaclav Kotesovec_, Sep 06 2013
%e L.g.f.: L(x) = 2*x + 4*x^2/2 + 20*x^3/3 + 296*x^4/4 + 10067*x^5/5 +...
%e where
%e exp(L(x)) = 1 + 2*x + 4*x^2 + 12*x^3 + 94*x^4 + 2195*x^5 + 158904*x^6 + 31681195*x^7 +...+ A228809(n)*x^n +...
%t Table[Sum[Binomial[n*k, k^2],{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Sep 06 2013 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n*k,k^2))
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A228809, A207136, A167009, A228832.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 04 2013
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