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A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k*+1) * n/(n-k).

%I A181081 #7 Apr 05 2021 00:06:42
%S A181081 1,3,7,39,336,4077,68461,1955295,129385141,17371664728,3501431925168,
%T A181081 947675920795833,355261887514210899,236156938257380344851,
%U A181081 390707976511340699319417,1324768245535417597286345871
%N A181081 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n-2k*+1) * n/(n-k).
%H A181081 G. C. Greubel, <a href="/A181081/b181081.txt">Table of n, a(n) for n = 1..90</a>
%F A181081 L.g.f.: L(x) = Sum_{n>=1} ( Sum_{k=0..n} binomial(n,k)^(n-k+1)*x^k ) * x^n/n.
%F A181081 Logarithmic derivative of A181080.
%e A181081 L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 39*x^4/4 + 336*x^5/5 + ...
%e A181081 which equals the series:
%e A181081 log(A(x)) = (1 + x)*x + (1 + 2^2*x + x^2)*x^2/2
%e A181081 + (1+ 3^3*x + 3^2*x^2 + x^3)*x^3/3
%e A181081 + (1+ 4^4*x + 6^3*x^2 + 4^2*x^3 + x^4)*x^4/4
%e A181081 + (1+ 5^5*x + 10^4*x^2 + 10^3*x^3 + 5^2*x^4 + x^5)*x^5/5
%e A181081 + (1+ 6^6*x + 15^5*x^2 + 20^4*x^3 + 15^3*x^4 + 6^2*x^5 + x^6)*x^6/6 + ...
%e A181081 Exponentiation yields the g.f. of A181080:
%e A181081 exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 14*x^4 + 83*x^5 + 774*x^6 + 10641*x^7 + ...
%t A181081 Table[Sum[Binomial[n-k, k]^(n-2*k+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* _G. C. Greubel_, Apr 04 2021 *)
%o A181081 (PARI) a(n)=sum(k=0, n\2, binomial(n-k, k)^(n-2*k+1)*n/(n-k))
%o A181081 (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, m, binomial(m,k)^(m-k+1)*x^k)*x^m/m)+x*O(x^n), n)}
%o A181081 (Magma) [(&+[Binomial(n-j,j)^(n-2*j+1)*(n/(n-j)): j in [0..Floor(n/2)]]): n in [1..20]]; // _G. C. Greubel_, Apr 04 2021
%o A181081 (Sage) [sum( binomial(n-k, k)^(n-2*k+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # _G. C. Greubel_, Apr 04 2021
%Y A181081 Cf. A181080 (exp), variants: A181071, A166895.
%K A181081 nonn
%O A181081 1,2
%A A181081 _Paul D. Hanna_, Oct 02 2010