cp's OEIS Frontend

A219267 Logarithmic derivative of the hyperfactorials (A002109).

%I A219267 #9 Jul 10 2015 07:18:33
%S A219267 1,7,313,110143,431860201,24185951471887,23238336572015738041,
%T A219267 445571476975584446962639039,194201470505208674769594891331807753,
%U A219267 2157794122078406207016487628429579826176795887,677208230450612019931822374477208301572175793625037599321
%N A219267 Logarithmic derivative of the hyperfactorials (A002109).
%C A219267 Hyperfactorial A002109(n) = Product_{k=0..n} k^k.
%D A219267 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.
%F A219267 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
%e A219267 L.g.f.: L(x) = x + 7*x^2/2 + 313*x^3/3 + 110143*x^4/4 + 431860201*x^5/5 +...
%e A219267 where
%e A219267 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 +...
%t A219267 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 *)
%o A219267 (PARI) {a(n)=n*polcoeff(log(sum(k=0,n+1,prod(j=0,k,j^j)*x^k)+x*O(x^n)),n)}
%o A219267 for(n=1,21,print1(a(n),", "))
%Y A219267 Cf. A002109, A219266, A219268.
%K A219267 nonn
%O A219267 1,2
%A A219267 _Paul D. Hanna_, Nov 16 2012