A193440 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).
1, 1, 2, 9, 145, 10489, 4182481, 10893144241, 213590500341121, 35762619247862532481, 57146369032805384396332801, 963199581177063129894232882156801, 187554502919537918586035198740350553881601, 458564976873147078680542618033293809080455988300801
Offset: 0
Keywords
Examples
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) +... where 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) +... and G(n) = 0!*1!*2!*3!*...*(n-1)!*n!.
Links
- Eric Weisstein's World of Mathematics, Barnes G-Function.
Crossrefs
Cf. A000178.
Programs
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Mathematica
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 *) -
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)}
Extensions
Definition corrected by Vaclav Kotesovec, Apr 03 2021
Comments