cp's OEIS Frontend

A181077 a(n) = Sum_{k=0..n-1} C(n-1,k)^(n-k) * n/(n-k).

%I A181077 #5 Apr 05 2021 20:40:38
%S A181077 1,3,10,59,726,20832,1405566,202357171,66675848266,52415395776938,
%T A181077 88820554918533846,339849475991699902472,3175234567292428864024192,
%U A181077 65420235446121559438182151848,2970041251569931717805628420162750
%N A181077 a(n) = Sum_{k=0..n-1} C(n-1,k)^(n-k) * n/(n-k).
%H A181077 G. C. Greubel, <a href="/A181077/b181077.txt">Table of n, a(n) for n = 1..75</a>
%F A181077 L.g.f.: L(x) = Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^n *x^k ] *x^n/n.
%F A181077 Logarithmic derivative of A181076.
%e A181077 L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 59*x^4/4 + 726*x^5/5 +...
%e A181077 which equals the series:
%e A181077   L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
%e A181077   + (1 + 2^2*x +  3^2*x^2 +  4^2*x^3 +   5^2*x^4 +   6^2*x^5 + ...)*x^2/2
%e A181077   + (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
%e A181077   + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 +  35^4*x^4 +  56^4*x^5 + ...)*x^4/4
%e A181077   + (1 + 5^5*x + 15^5*x^2 + 35^5*x^3 +  70^5*x^4 + 126^5*x^5 + ...)*x^5/5
%e A181077   + (1 + 6^6*x + 21^6*x^2 + 56^6*x^3 + 126^6*x^4 + 252^6*x^5 + ...)*x^6/6
%e A181077   + (1 + 7^7*x + 28^7*x^2 + 84^7*x^3 + 210^7*x^4 + 462^7*x^5 + ...)*x^7/7 + ...
%e A181077 Exponentiation yields the g.f. of A181076:
%e A181077   exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 20*x^4 + 168*x^5 + 3659*x^6 + ... + A181076(n)*x^n + ...
%t A181077 Table[Sum[Binomial[n-1, k]^(n-k)*n/(n-k), {k,0,n-1}], {n,25}] (* _G. C. Greubel_, Apr 05 2021 *)
%o A181077 (PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-k)*n/(n-k))}
%o A181077 (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^m*x^k)*x^m/m)+x*O(x^n), n)}
%o A181077 (Magma) [(&+[Binomial(n-1, k)^(n-k)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // _G. C. Greubel_, Apr 05 2021
%o A181077 (Sage) [sum(binomial(n-1,k)^(n-k)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # _G. C. Greubel_, Apr 05 2021
%Y A181077 Cf. A181076 (exp), variants: A181073, A181075, A181079.
%K A181077 nonn
%O A181077 1,2
%A A181077 _Paul D. Hanna_, Oct 02 2010