When I was examining the product,
(1) $\prod _{i=1}^{\infty } H_{i} ^{(i)}$
I noticed that product converges to a single constant, 1.6798, and does so quite quickly. Is this constant something special, or is a special value of another function, or is it just a combination of other known constants? I'm using this product as part of an investigation of differential equations of form,
(2) $y'(x)=\prod _{i=1}^n (H_i^{(i)} x^{n-i} \space y(x)^i ) = x^{\frac{1}{2} (n-1) n} \space y(x)^{\frac{1}{2} n (n+1)}\prod _{i=1}^{\infty } H_{i} ^{(i)}$,
where the product (1) above is the limiting constant of the product (2).
A few more digits $$ 1.6798002778544903357 $$ (where the last digit shown has rounded up). The Inverse Symbolic Calculator doesn't recognize it, so you're probably out of luck.