A219266 Logarithmic derivative of the superfactorials (A000178).

1, 3, 31, 1103, 171311, 149089887, 877704854447, 40451674467223423, 16514355739866259408591, 66586047491662065505372477983, 2923692867015618804999172694908629103, 1527767556403309713534536695030930443376591295, 10306227067090276816548435451550663056418226402352755215

Offset: 1

Author

Paul D. Hanna, Nov 16 2012

Keywords

• nonn

Comments

Superfactorial A000178(n) equals the product of first n factorials.

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 31*x^3/3 + 1103*x^4/4 + 171311*x^5/5 +...
where
exp(L(x)) = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + 125411328000*x^7 +...+ n!*(n-1)!*(n-2)!*...*3!*2!*1!*0!**x^n +...
		

Crossrefs

Cf. A000178, A219267, A219268, A219269.

Programs

Mathematica

nmax=15; Rest[CoefficientList[Series[Log[Sum[BarnesG[k+2]*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!)*x^k)+x*O(x^n)),n)}
for(n=1,21,print1(a(n),", "))

Formula

a(n) ~ n^(n^2/2 + n + 17/12) * (2*Pi)^((n+1)/2) / (A * exp(3*n^2/4 + n - 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015