Consider a general template to generate sequences (or polynomials) using the inverse Mellin transform and a kernel function ϕ(s) $$ p_k(x) = f(x) ^{-1}[\phi(s) q_k(s)](x) $$ here pk(x) and qk(x) are polynomials, and f(x) is a function that cancels out with the generating form from the inverse Mellin transform. This is observed with an example setting qk(s)=sk, ϕ(s)=Γ(s) and f(x)=ex, we have $$ B'_k(x) = e^x ^{-1}[\Gamma(s) s^k](x) $$ where B′k(x) appear to be some form of alternating Bell polynomials, and the coefficients of these polynomials are made up of Stirling numbers of the second kind S₂(n, k) as $$ B'_n(x) = ^n (-1)^{n-k} S_2(n,k) x^k $$ we also find that $$ ^n S_2(n,k)}{2^n} x^{k/2} = e^{} ^{-1}[\Gamma(2s) s^n](x) $$ very interestingly $$ (1+x)^{n+1} ^{-1}[\Gamma(s)\Gamma(1-s) s^n](x) ^n (-1)^{n-k-1} A[n,k] x^{k+1}, k>0 $$ where A(n, k) as the Eulerian numbers. The agreement is off slightly for k = 0. There is a more general form to this $$ (1+x)^{n+t} ^{-1}[{\Gamma(t)} s^n](x) $$ which for t = 1 gives the Eulerian numbers, and for t = 2 is related to A199335. We can even insert t = 1/2, and get a sequence which is related to A185411 (with an additional factor to 1/2n). FIXING THE SIGNS We now consider a modification to the transform to fix the signs, define the inverse-Q transform as $$ p_n(x) = ^{-1}[\phi(s)](n,x) = ^{-1}[\phi(s) (-s)^n](-x) $$ where we have chosen the inverse because of the inverse Mellin transforms, now we have $$ ^{-1}[\Gamma(s)](x) ^{-1}[\Gamma(s)](n,x) = B_n(x) = ^n S_2(n,k) x^k $$ for Bell polynomials Bn(x) and interpreting 0⁰ as 1 which is common in combinatorics. It’s still (perhaps) not entirely right, because for ϕ(s)=Γ(s)Γ(1 − s) we have $$ (1-x)^{n+1}^{-1}[\Gamma(s)\Gamma(1-s)](n,x) = x A_n(x), n>0 $$ relating to Eularian polynomials, equally one could say $$ (1-x)^{-1}[\Gamma(s)\Gamma(1-s)](x) = {(1-x)^n}, n>0 $$ TABLE OF RELATIONS |c|c| Function & Function & Numbers Γ(s) & ex & StirlingS2 Γ(s)Γ(1 − s) & (1 + x)n + 1 & Eulerian Numbers we can see that the function f(x) is clearly related to ℳ−1[ϕ(s)], which is exciting because, by assuming qk(s)=sk for all inputs it links the function ϕ(s) directly a special class of numbers T(n, k). We can as questiosn such as , which kernel ϕ(s) produces the binomials?