Review a preprint named Johann Faulhaber and Sums of Powers by Donald E. Knuth, on page 10 are shown sums of $n$ to odd powers, for example, $$\sum n^1=\binom{n+1}{2},$$ $$\sum n^3=6\binom{n+2}{4}+\binom{n+1}{2},$$ $$\sum n^5=120\binom{n+3}{6}+30\binom{n+2}{4}+\binom{n+1}{2},$$
and so on...
Question 1. How can we produce coefficients $\{1,6,1,120,30,1\}$ in this expressions by generating function, and what function is it?
PS they are closely related to central factorial numbers https://oeis.org/A008957 but exact formula is states unclear.
Your Question 1 is answered in OES sequence A008957. For example, the coefficients $\;(1,6,1,120,30,1,\dots)\;$ are expressed in terms of A008957 exactly how Knuth meant it: $$\;(1!\cdot1,\;\;3!\cdot1,1!\cdot1,\;\;5!\cdot1,3!\cdot5,1!\cdot1,\;\;7!\cdot1,5!\cdot14,3!\cdot21,1!\cdot1,\;\;\dots).\;$$ The triangular sequence A008957 has as exponential generating function $$ x^2\! \cosh(\sinh(y\; x/2)/(x/2))\!-\!1)= (1x^2)y^2/2!\! +\!(1x^2\!+\!1x^4)y^4/4!\! +\!(1x^2\!+\!5x^4\!+\!1x^6)y^6/6!+\cdots.$$