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.
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
KEYWORD
nonn
AUTHOR
Thomas Scheuerle, Aug 04 2022
STATUS
approved