A355129 a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).

2, 3, 7, 40, 856, 91821, 60080136, 279276911843, 10503211888973754, 3585680755683196123365, 12323227994417456429490342865, 468378989392773003347310901356953089, 214565221409985003242070442557341938941878313, 1282499669290042152350268651085002913530161723080398635

Offset: 0

Author

Thomas Scheuerle, Aug 04 2022

Keywords

• nonn

Comments

List of the possible cases regarding the patterns of the numbers in the sequence b:

Length: 1 2 3 4 5 6

Pos 0: 1 1 1 1 1 1

Pos 1: 1 2 3 4 5 6

Pos 2: 0 0 3 7 12 18

Pos 3: 0 0 0 7 19 37

Pos 4: 0 0 0 7 26 63

Pos 5: 0 0 0 7 33 96

Pos 6: 0 0 0 7 40 136

Pos 7: 0 0 0 0 40 176

Pos 8: 0 0 0 0 40 216

... ... ... ... ... ... ...

Sum: 2 3 7 40 856 91821

Each row counts the number of possible distributions of numbers, row "Pos 0" is the number of possible distributions with only the number zero. The row "Pos 1" counts the distributions of zeros and ones. The row "Pos 2" the possible distributions of {0,1,2} and so forth.

From top to down: If a number in the column length = k has reached the value of the sum of the column length = k-1, this number will be k!-(k-1)!+1 times repeated. Before this limit is reached each number is the sum of the neighbor one step above and the neighbor one step to the left.

Examples

			For a(0) we get two possible sequences:
  {0}, {1}.
For a(1) we get three possible sequences:
  {0, 0}, {0, 1}, {1, 1}.
For a(2) = 7 we get:
  {0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},
  {0, 1, 2}, {1, 1, 1}, {1, 1, 2}.
		

Crossrefs

Cf. A000108 (if we change the definition into 0 <= b(k) <= k).

Cf. A005269, A005270, A008934, A016121, A128094, A242105.

Programs

PARI

a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + sum(r = 1, n-2, sum(k = 0, r-1 ,binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1)))

Formula

a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + Sum_{r = 1..n-2} Sum_{k = 0..r-1} binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1).