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a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.
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%I #14 Nov 20 2024 08:08:31

%S 1,3,7,39,366,5697,194881,16288695,2430565261,564615230758,

%T 257227244037248,319346787227133873,832952161388710135215,

%U 3382434539389226013260403,22966972221673234523620345857

%N a(n) = Sum_{k=0..[n/2]} C(n-k,k)^(n-k)*n/(n-k), n>=1.

%H G. C. Greubel, <a href="/A166895/b166895.txt">Table of n, a(n) for n = 1..50</a>

%F Logarithmic derivative of A166894.

%F Limit_{n->oo} a(n)^(1/n^2) = (1/r - 1)^((1 - r)^2/(3 - 4*r)) = 1.436094496902535711953511352318447104797138641971237143543..., where r = A323777 = 0.220676041323740696312822269998050167187681031027574... is the root of the equation (1 - 2*r)^(3 - 4*r) = (1 - r)^(2*(1 - r))*r^(1 - 2*r). - _Vaclav Kotesovec_, Nov 20 2024

%e L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 366*x^5/5 + 5697*x^6/6 +...

%e exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 89*x^5 + 1050*x^6 +...+ A166894(n)*x^n +...

%t Table[Sum[Binomial[n - k, k]^(n - k) *n/(n - k), {k, 0, Floor[n/2]}], {n, 1, 25}] (* _G. C. Greubel_, May 27 2016 *)

%o (PARI) a(n)=sum(k=0,n\2,binomial(n-k,k)^(n-k)*n/(n-k))

%Y Cf. A166894, A209428.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 23 2009