OFFSET
0,2
COMMENTS
Adjacent terms of s(n, m) from the formula section (for m > 1) have k identical digits at the end in any number system q > 1.
FORMULA
b(2n+1, m) = m*b(n, m) for n >= 0.
b(2n, m) = b(n, m) + b(n - 2^f(n), m) + b(2n - 2^f(n), m) for n > 0 with b(0, m) = 1 where f(n) = A007814(n).
s(n, m) = Sum_{k=0..2^n - 1} b(k, m) = (n+m)*s(n-1, m) + ((m+1)^2 - 4)*(n+m-1)*(g(n+m-2) - g(m+1))/((m+3)*(m+1)!) for n > 0 with s(0, m) = 1 where g(n) = A003422(n).
a(n) = s(n, 2) for n >= 0.
PROG
(PARI) lf(n) = sum(k=0, n-1, k!); \\ A003422
a(n) = if (n, (n+2)*a(n-1) + (n+1)*(lf(n) - 4)/6, 1); \\ Michel Marcus, Jan 11 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Mikhail Kurkov, Dec 24 2021 [verification needed]
STATUS
approved