A181075 a(n) = Sum_{k=0..n-1} C(n-1,k)^(k+1) * n/(n-k).

1, 3, 10, 71, 1026, 30912, 2219946, 339460991, 112986526834, 91234232847938, 161113616883239406, 619495336824891912596, 5839092706931985694730356, 124192664709851995516427897172, 5681764626723349386531457243004370

Offset: 1

Author

Paul D. Hanna, Oct 02 2010

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 71*x^4/4 + 1026*x^5/5 + ...
which equals the series:
  L(x) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
  + (1 + 2^2*x +  3^3*x^2 +  4^4*x^3 +   5^5*x^4 +   6^6*x^5 + ...)*x^2/2
  + (1 + 3^2*x +  6^3*x^2 + 10^4*x^3 +  15^5*x^4 +  21^6*x^5 + ...)*x^3/3
  + (1 + 4^2*x + 10^3*x^2 + 20^4*x^3 +  35^5*x^4 +  56^6*x^5 + ...)*x^4/4
  + (1 + 5^2*x + 15^3*x^2 + 35^4*x^3 +  70^5*x^4 + 126^6*x^5 + ...)*x^5/5
  + (1 + 6^2*x + 21^3*x^2 + 56^4*x^3 + 126^5*x^4 + 252^6*x^5 + ...)*x^6/6
  + (1 + 7^2*x + 28^3*x^2 + 84^4*x^3 + 210^5*x^4 + 462^6*x^5 + ...)*x^7/7 + ...
Exponentiation yields the g.f. of A181074:
  exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 23*x^4 + 231*x^5 + 5405*x^6 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..75

Crossrefs

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

Programs

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

Mathematica

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

PARI

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

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)}

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

Formula

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

Logarithmic derivative of A181074.