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

0,1

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.

Links

Table of n, a(n) for n=0..13.

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).

Example

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}.

Prog

(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)))

Crossrefs

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

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

Sequence in context: A182219 A227777 A037843 * A102604 A119662 A265776

Adjacent sequences: A355126 A355127 A355128 * A355130 A355131 A355132

Keyword

nonn

Author

Thomas Scheuerle, Aug 04 2022

Status

approved