|
%I #13 Apr 03 2021 09:43:59
%S 1,1,2,9,145,10489,4182481,10893144241,213590500341121,
%T 35762619247862532481,57146369032805384396332801,
%U 963199581177063129894232882156801,187554502919537918586035198740350553881601,458564976873147078680542618033293809080455988300801
%N exp( Sum_{n>=1} x^n/G(n) ) = Sum_{n>=0} a(n)*x^n/G(n), where G(n) = Product_{k=0..n} k! = BarnesG(n+2), (see A000178).
%C Sum_{n>=0} a(n)/G(n) = 4.88825080515459884947818345139584332...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%e A(x) = 1 + x + 2*x^2/2 + 9*x^3/12 + 145*x^4/288 + 10489*x^5/34560 + 4182481*x^6/24883200 + 10893144241*x^7/125411328000 +...+ a(n)*x^n/G(n) +...
%e where
%e log(A(x)) = x + x^2/2 + x^3/12 + x^4/288 + x^5/34560 + x^6/24883200 + x^7/125411328000 +...+ x^n/G(n) +...
%e and G(n) = 0!*1!*2!*3!*...*(n-1)!*n!.
%t Table[BarnesG[n+2] * SeriesCoefficient[Exp[Sum[x^k/BarnesG[k+2], {k, 1, n}]], {x, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Apr 03 2021 *)
%o (PARI) {a(n)=prod(k=1,n,k!)*polcoeff(exp(sum(m=1,n+1,x^m/prod(k=1,m,k!)+x*O(x^n))),n)}
%Y Cf. A000178.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 25 2011
%E Definition corrected by _Vaclav Kotesovec_, Apr 03 2021
|
