A181079 - OEIS

%I #9 Apr 05 2021 00:06:33

%S 1,3,10,95,3126,363132,154742736,238830058287,1401973344195850,

%T 30168336369959767298,2525043541826640689536056,

%U 779938173975597096091742711900,951131113887078985926203597341181404

%N a(n) = Sum_{k=0..n-1} binomial(n-1,k)^(n-1) * n/(n-k).

%H G. C. Greubel, <a href="/A181079/b181079.txt">Table of n, a(n) for n = 1..60</a>

%F L.g.f.: L(x) = Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^(n+k-1)*x^k ] *x^n/n.

%F Logarithmic derivative of A181076.

%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 95*x^4/4 + 3126*x^5/5 + ...

%e which equals the series:

%e L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + ...)*x

%e + (1 + 2^2*x + 3^3*x^2 + 4^4*x^3 + 5^5*x^4 + 6^6*x^5 + ...)*x^2/2

%e + (1 + 3^3*x + 6^4*x^2 + 10^5*x^3 + 15^6*x^4 + 21^7*x^5 + ...)*x^3/3

%e + (1 + 4^4*x + 10^5*x^2 + 20^6*x^3 + 35^7*x^4 + 56^8*x^5 + ...)*x^4/4

%e + (1 + 5^5*x + 15^6*x^2 + 35^7*x^3 + 70^8*x^4 + 126^9*x^5 + ...)*x^5/5

%e + (1 + 6^6*x + 21^7*x^2 + 56^8*x^3 + 126^9*x^4 + 252^10*x^5 + ...)*x^6/6

%e + (1 + 7^7*x + 28^8*x^2 + 84^9*x^3 + 210^10*x^4 + 462^11*x^5 + ...)*x^7/7 + ...

%e Exponentiation yields the g.f. of A181078:

%e exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 29*x^4 + 657*x^5 + 61207*x^6 + … + A181078(n)*x^n + ...

%t Table[Sum[Binomial[n-1,k]^(n-1) n/(n-k),{k,0,n-1}],{n,20}] (* _Harvey P. Dale_, Jun 13 2013 *)

%o (PARI) {a(n)=sum(k=0, n-1, binomial(n-1, k)^(n-1)*n/(n-k))}

%o (PARI) {a(n)=n*polcoeff(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^(m+k-1)*x^k)*x^m/m)+x*O(x^n), n)}

%o (Magma) [(&+[Binomial(n-1,j)^(n-1)*(n/(n-j)): j in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Apr 04 2021

%o (Sage) [sum( binomial(n-1, k)^(n-1)*(n/(n-k)) for k in (0..n-1)) for n in (1..20)] # _G. C. Greubel_, Apr 04 2021

%Y Cf. A181078 (exp), variants: A181071, A181075, A181077.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 03 2010