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A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).

%I A181083 #10 Apr 06 2021 04:52:04
%S A181083 1,3,13,111,1686,88737,14355265,3583775847,1789371713317,
%T A181083 4311992850152298,23667113846872049808,185391762466214524964649,
%U A181083 4305238471804328835068596175,468653724243371951619336632177235
%N A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).
%H A181083 G. C. Greubel, <a href="/A181083/b181083.txt">Table of n, a(n) for n = 1..70</a>
%F A181083 L.g.f.: L(x) = Sum_{n>=1} [Sum_{k=0..n} binomial(n,k)^(n+k)*x^k] * x^n/n.
%F A181083 Logarithmic derivative of A181082.
%e A181083 L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + ...
%e A181083 which equals the series:
%e A181083   L(x) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
%e A181083   + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
%e A181083   + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
%e A181083   + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
%e A181083   + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...
%e A181083 Exponentiation yields the g.f. of A181082:
%e A181083   exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 + 2066401*x^7 + ...
%p A181083 A181083:= n-> add(binomial(n-k,k)^n*(n/(n-k)), k=0..floor(n/2)); seq(A181083(n), n=1..20); # _G. C. Greubel_, Apr 05 2021
%t A181083 Table[Sum[Binomial[n-k,k]^n n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* _Harvey P. Dale_, Jun 24 2015 *)
%o A181083 (PARI) a(n)=sum(k=0, n\2, binomial(n-k, k)^n*n/(n-k))
%o A181083 (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m+k)*x^k)*x^m/m)+x*O(x^n), n)}
%o A181083 (Magma) [(&+[Binomial(n-k,k)^n*(n/(n-k)): k in [0..Floor(n/2)]]): n in [1..25]]; // _G. C. Greubel_, Apr 05 2021
%o A181083 (Sage) [sum(binomial(n-k,k)^n*(n/(n-k)) for k in (0..n//2)) for n in (1..25)] # _G. C. Greubel_, Apr 05 2021
%Y A181083 Variants: A166895, A181071, A181081.
%Y A181083 Cf. A181082 (exp).
%K A181083 nonn
%O A181083 1,2
%A A181083 _Paul D. Hanna_, Oct 02 2010