The terms at x=0 define the Bernoulli twin numbers, C(n,0)=C(n) = A129826(n)/(n+1)! .
Because the C(n,x) are derived from the Bernoulli polynomials B(n,x) via a binomial transformation and because the odd-indexed Bernoulli numbers are (essentially) zero, the following sum rules for the C(n) emerge (partially in Umbral notation):
For odd C(n): C(2n)=(C-1)^(2n-1), n > 1, C(2n) disappears; example: C(4)=C(4)-3C(3)+3C(2)-C(1).
0r for C(2n+1): (C-1)^2n=0, n >0; example: C(1)-4C(2)+6C(3)-4C(4)+C(5)=0.
With positive coefficients, table
1, 2;
2, 2, 3;
3, 2, 3, 6;
4, 2, 3, 6, 30;
5, 2, 3, 6, 30, -30;
6, 2, 3, 6, 30, -30, -42;
gives C(n). Example: 3C(0)+2C(1)+3C(2)+6C(3)=0. See -A051717(n+1), Bernoulli twin numbers denominators, with from 30 opposite twin.
