A219267 Logarithmic derivative of the hyperfactorials (A002109).

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

Offset

1,2

Comments

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

References

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.

Links

Table of n, a(n) for n=1..11.

Formula

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

Example

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 +...

Mathematica

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 *)

Prog

(PARI) {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), ", "))

Crossrefs

Cf. A002109, A219266, A219268.

Sequence in context: A119163 A364479 A171148 * A244851 A082160 A220278

Adjacent sequences: A219264 A219265 A219266 * A219268 A219269 A219270

Keyword

nonn

Author

Paul D. Hanna, Nov 16 2012

Status

approved