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Dear Experts!

If we check the limits of the ratio of the gamma function and the 3rd and 4th hyperoperators (exponentiation and tetration), then we see the the gamma function should be an interoperator. I show it:
lim exp(h) / h! = 0
lim h! / h^^2 = 0
lim h!^(1/h) / h = 1/e
lim log(h!)/log(h) / h = 1
h->+oo
I guess the level of iteration of an analytic increasing function, operator depends on its limits at infinity as also its asymptote at infinity. It is important to investigate the higher- and interhyperoperators. Now, I would like to generate the functional inverse of the gamma.
I have used matrices Carleman.

Code:
Dfac(x,N,h)=intnum(t=0,h,t^x/exp(t)*log(t)^N)
/*Nth derivative of function gamma*/
n=11
A=matrix(n,n,i,j,Dfac(0.0,j,1000.0)^i/j!)
defac(x)=sum(k=0,n,(A^-1)[1,k]*x^k)

I checked if defac(24)=4, but no, it does not work.
Please, help me.

Best regards,
Xorter Unizo