A181075 - OEIS

%I #5 Apr 05 2021 20:40:29

%S 1,3,10,71,1026,30912,2219946,339460991,112986526834,91234232847938,

%T 161113616883239406,619495336824891912596,5839092706931985694730356,

%U 124192664709851995516427897172,5681764626723349386531457243004370

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

%H G. C. Greubel, <a href="/A181075/b181075.txt">Table of n, a(n) for n = 1..75</a>

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

%F Logarithmic derivative of A181074.

%e L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*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 + 3^2*x +  6^3*x^2 + 10^4*x^3 +  15^5*x^4 +  21^6*x^5 + ...)*x^3/3

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

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

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

%e   + (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...

%e Exponentiation yields the g.f. of A181074:

%e   exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 + ...

%t Table[Sum[Binomial[n-1, k]^(k+1)*n/(n-k), {k,0,n-1}], {n,25}] (* _G. C. Greubel_, Apr 05 2021 *)

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

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

%o (Magma) [(&+[Binomial(n-1, k)^(k+1)*n/(n-k): k in [0..n-1]]): n in [1..25]]; // _G. C. Greubel_, Apr 05 2021

%o (Sage) [sum(binomial(n-1,k)^(k+1)*n/(n-k) for k in (0..n-1)) for n in (1..25)] # _G. C. Greubel_, Apr 05 2021

%Y Cf. A181076 (exp), variants: A181077, A181079.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 02 2010