A138734 a(n) = n-th term of the (n+1)-th inverse binomial transform of A138737 for n>=0.

1, 0, 4, 0, 176, 0, 24192, 0, 6966528, 0, 3459768320, 0, 2642804895744, 0, 2880385956446208, 0, 4248075487709560832, 0, 8150896954121845211136, 0, 19738484719911211593891840, 0, 58891248341123034677673721856, 0

Offset: 0

Author

Paul D. Hanna, Apr 05 2008

Keywords

• nonn

Comments

The n-th term of the n-th inverse binomial transform of A138737 = (n+1)^(n-1) for n>=0.

Examples

			E.g.f.: A(x) = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! +...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A138737, A215880.

Programs

Mathematica

CoefficientList[Series[-LambertW[x]*LambertW[-x]/x^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 08 2013 *)

PARI

{a(n)=local(A=[1]);for(k=1,n,A=concat(A,0); A[k+1]=(k+1)^(k-1)-polcoeff(subst(Ser(A),x,x/(1+k*x+x*O(x^k)))/(1+k*x),k)); polcoeff(subst(Ser(A),x,x/(1+(n+1)*x+x*O(x^n)))/(1+(n+1)*x),n)}

PARI

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

PARI

{a(n)=local(LW=sum(m=1,n+1,m^(m-1)*x^m/m!)+x^2*O(x^n));n!*polcoeff(-LW*subst(LW,x,-x)/x^2,n)}

PARI

{a(n)=n!*polcoeff(exp(sum(m=1,n,2*(2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)),n)}

PARI

x='x+O('x^50); Vec(serlaplace(-lambertw(x)*lambertw(-x)/x^2)) \\ G. C. Greubel, Nov 15 2017

Formula

From Paul D. Hanna, Apr 20 2011: (Start)

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

E.g.f.: -LambertW(x)*LambertW(-x)/x^2.

E.g.f.: exp( Sum_{n>=1} 2*(2n)^(2n-1) * x^(2n)/(2n)! ). (End)

Let Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)!

and Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!

then e.g.f. A(x) = Chw(x)^2 - Shw(x)^2. - Paul D. Hanna, Aug 22 2012

If n is even, a(n) ~ 2*exp(2) * LambertW(exp(-1)) * n^(n-1). - Vaclav Kotesovec, Oct 08 2013