Consider Pascal's triangle without the $1$s.
Let $S(n)$ be the sum of $n$th powers of the reciprocals, where $n$ is an integer greater than $1$. ($S(1)$ diverges because the harmonic series diverges.)
$$S(n)=\frac{1}{2^n}+\frac{1}{3^n}+\frac{1}{3^n}+\frac{1}{4^n}+\frac{1}{6^n}+\frac{1}{4^n}+\frac{1}{5^n}+\cdots$$
What is a closed form for $S(n)$ in terms of $n$?
Earlier we found that $S(2)=\frac13+\frac{4}{3\sqrt 3}G\approx 1.11463574623$ where $G$ is Gieseking's constant. This question seeks to generalize this result.
Using the first 150 rows, here are rough approximations:
$S(2)\approx1.101346$
$S(3)\approx0.286888$
$S(4)\approx0.103181$
$S(5)\approx0.042758$
$S(6)\approx0.019085$
$S(7)\approx0.008890$
$S(8)\approx0.004249$
$S(9)\approx0.002064$
$S(10)\approx0.001013$
Wolfram does not evaluate $S(n)$.
