A181083 a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^n * n/(n-k).

1, 3, 13, 111, 1686, 88737, 14355265, 3583775847, 1789371713317, 4311992850152298, 23667113846872049808, 185391762466214524964649, 4305238471804328835068596175, 468653724243371951619336632177235

Offset: 1

Author

Paul D. Hanna, Oct 02 2010

Keywords

• nonn

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 13*x^3/3 + 111*x^4/4 + 1686*x^5/5 + ...
which equals the series:
  L(x) = (1 + x)*x + (1 + 2^3*x + x^2)*x^2/2
  + (1 + 3^4*x +  3^5*x^2 +      x^3)*x^3/3
  + (1 + 4^5*x +  6^6*x^2 +  4^7*x^3 +       x^4)*x^4/4
  + (1 + 5^6*x + 10^7*x^2 + 10^8*x^3 +   5^9*x^4 +      x^5)*x^5/5
  + (1 + 6^7*x + 15^8*x^2 + 20^9*x^3 + 15^10*x^4 + 6^11*x^5 + x^6)*x^6/6 + ...
Exponentiation yields the g.f. of A181082:
  exp(L(x)) = 1 + x + 2*x^2 + 6*x^3 + 34*x^4 + 375*x^5 + 15200*x^6 + 2066401*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..70

Crossrefs

Variants: A166895, A181071, A181081.

Cf. A181082 (exp).

Programs

Magma

[(&+[Binomial(n-k,k)^n*(n/(n-k)): k in [0..Floor(n/2)]]): n in [1..25]]; // G. C. Greubel, Apr 05 2021

Maple

A181083:= n-> add(binomial(n-k,k)^n*(n/(n-k)), k=0..floor(n/2)); seq(A181083(n), n=1..20); # G. C. Greubel, Apr 05 2021

Mathematica

Table[Sum[Binomial[n-k,k]^n n/(n-k),{k,0,Floor[n/2]}],{n,20}] (* Harvey P. Dale, Jun 24 2015 *)

PARI

a(n)=sum(k=0, n\2, binomial(n-k, k)^n*n/(n-k))

PARI

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

Sage

[sum(binomial(n-k,k)^n*(n/(n-k)) for k in (0..n//2)) for n in (1..25)] # G. C. Greubel, Apr 05 2021

Formula

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

Logarithmic derivative of A181082.