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A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).

%I A181085 #7 Apr 05 2021 00:06:55
%S A181085 1,3,25,327,6336,513657,142074241,52903930911,36806786795365,
%T A181085 148308705637730728,1318954828711012426638,15279013243159345043036553,
%U A181085 534104982404807772659968455891,97749134742042348389685885848315523
%N A181085 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^(n+1) * n/(n-k).
%H A181085 G. C. Greubel, <a href="/A181085/b181085.txt">Table of n, a(n) for n = 1..60</a>
%e A181085 L.g.f.: L(x) = x + 3*x^2/2 + 25*x^3/3 + 327*x^4/4 + 6336*x^5/5 + ...
%e A181085 which equals the series:
%e A181085 L(x) = (1 + x)*x + (1 + 2^4*x + x^2)*x^2/2
%e A181085 + (1+ 3^5*x + 3^6*x^2 + x^3)*x^3/3
%e A181085 + (1+ 4^6*x + 6^7*x^2 + 4^8*x^3 + x^4)*x^4/4
%e A181085 + (1+ 5^7*x + 10^8*x^2 + 10^9*x^3 + 5^10*x^4 + x^5)*x^5/5
%e A181085 + (1+ 6^8*x + 15^9*x^2 + 20^10*x^3 + 15^11*x^4 + 6^12*x^5 + x^6)*x^6/6 + ...
%e A181085 Exponentiation yields the g.f. of A181084:
%e A181085 exp(L(x)) = 1 + x + 2*x^2 + 10*x^3 + 92*x^4 + 1367*x^5 + 87090*x^6 + ...
%t A181085 Table[Sum[Binomial[n-k, k]^(n+1)*(n/(n-k)), {k, 0, Floor[n/2]}], {n, 20}] (* _G. C. Greubel_, Apr 04 2021 *)
%o A181085 (PARI) a(n)=sum(k=0, n\2, binomial(n-k, k)^(n+1)*n/(n-k))
%o A181085 (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 A181085 (Sage) [sum( binomial(n-k, k)^(n+1)*(n/(n-k)) for k in (0..n//2)) for n in (1..20)] # _G. C. Greubel_, Apr 04 2021
%o A181085 (Magma) [(&+[Binomial(n-j,j)^(n+1)*(n/(n-j)): j in [0..Floor(n/2)]]): j in [1..20]]; // _G. C. Greubel_, Apr 04 2021
%Y A181085 Variants: A166895, A181071, A181081, A181083.
%Y A181085 Cf. A181084 (exp).
%K A181085 nonn
%O A181085 1,2
%A A181085 _Paul D. Hanna_, Oct 28 2010