A219268 Logarithmic derivative of A001142, where A001142(n) = product{k=1..n} k^k/k!.

1, 3, 22, 347, 11986, 956334, 184142134, 87903876147, 105736320973732, 323943204887363938, 2547547949361933790328, 51735228018482706470521574, 2726127372514537039881847535054, 374214400937086673452020875815709240, 134262616041282033840675468757467513112522

Offset: 1

Author

Paul D. Hanna, Nov 16 2012

Keywords

• nonn

Comments

A001142(n) = hyperfactorial(n)/superfactorial(n) = A002109(n)/A000178(n).

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 22*x^3/3 + 347*x^4/4 + 11986*x^5/5 + 956334*x^6/6 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 9*x^3 + 96*x^4 + 2500*x^5 + 162000*x^6 + 26471025*x^7 + 11014635520*x^8 +...+ A001142(n)*x^n +...
		

Crossrefs

Cf. A001142, A219266, A219268.

Programs

Mathematica

nmax=15; Rest[CoefficientList[Series[Log[Sum[Product[j^j/j!,{j,1,k}]*x^k,{k,0,nmax}]],{x,0,nmax}],x] * Range[0,nmax]] (* Vaclav Kotesovec, Jul 10 2015 *)

PARI

{a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j/j!)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ A^2 * exp(n^2/2 + n - 1/12) / (n^(n/2 - 2/3) * (2*Pi)^((n+1)/2)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015