A350309 a(n) = (n+2)*a(n-1) + (n+1)*(A003422(n) - 4)/6 for n > 0 with a(0) = 1.

1, 2, 7, 35, 215, 1535, 12455, 113255, 1141415, 12632615, 152341415, 1988514215, 27934434215, 420236744615, 6740662856615, 114841743944615, 2071122598472615, 39418302548552615, 789563088403016615, 16603426141551176615, 365724864314899016615, 8421063413387754056615

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.

Links

Table of n, a(n) for n=0..21.

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

Cf. A003422, A006157 (first differences), A007814.

Similar recurrences: A006014, A124758, A217924, A243499, A284005, A290615, A329369, A341392, A347204, A347205.

Sequence in context: A130458 A003575 A043546 * A307441 A260530 A201690

Adjacent sequences: A350306 A350307 A350308 * A350310 A350311 A350312

Keyword

nonn,changed

Author

Mikhail Kurkov, Dec 24 2021 [verification needed]

Status

approved