Recurrence for A000670

Question

A000670 contain a formula by Martin Kochanski:

Recurrence: $2a(n)=(a+1)^n$ where superscripts are converted to subscripts after binomial expansion - reminiscent of Bernoulli numbers $B_n=(B+1)^n$.

What does it mean?

Answer 1

It means that instead of $$(a+1)^n=\sum _{k=0}^n\binom{n}{k}a^k,$$ you use $$(a+1)_n=\sum _{k=0}^n\binom{n}{k}a_k$$