A219268 -id:A219268 - cp's OEIS frontend

A219266 Logarithmic derivative of the superfactorials (A000178).

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

Offset: 1

Paul D. Hanna, Nov 16 2012

nonn

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

			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 +...

Cf. A000178, A219267, A219268, A219269.

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 *)

{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),", "))

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

A219267 Logarithmic derivative of the hyperfactorials (A002109).

Original entry on oeis.org

1, 7, 313, 110143, 431860201, 24185951471887, 23238336572015738041, 445571476975584446962639039, 194201470505208674769594891331807753, 2157794122078406207016487628429579826176795887, 677208230450612019931822374477208301572175793625037599321

Offset: 1

Paul D. Hanna, Nov 16 2012

nonn

Hyperfactorial A002109(n) = Product_{k=0..n} k^k.

			L.g.f.: L(x) = x + 7*x^2/2 + 313*x^3/3 + 110143*x^4/4 + 431860201*x^5/5 +...
where
exp(L(x)) = 1 + x + 4*x^2 + 108*x^3 + 27648*x^4 + 86400000*x^5 + 4031078400000*x^6 +...+ n^n*(n-1)^(n-1)*(n-2)^(n-2)*...*3^3*2^2*1^1*0^0**x^n +...

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A002109, A219266, A219268.

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

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

a(n) ~ A * n^(n*(n+1)/2 + 13/12) / exp(n^2/4), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

cp's OEIS Frontend

A219266 Logarithmic derivative of the superfactorials (A000178).

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