A242105 -id:A242105 - cp's OEIS frontend

A261897 Triangle read by rows: T(n,k) (1 <= k <= n+1) = number of sequences of length n, dominated by the squares, with entries from [0,k] and largest entry k.

1, 1, 1, 0, 2, 1, 0, 2, 3, 1, 0, 2, 5, 4, 1, 0, 0, 7, 9, 5, 1, 0, 0, 7, 16, 14, 6, 1, 0, 0, 7, 23, 30, 20, 7, 1, 0, 0, 7, 30, 53, 50, 27, 8, 1, 0, 0, 7, 37, 83, 103, 77, 35, 9, 1, 0, 0, 0, 44, 120, 186, 180, 112, 44, 10, 1, 0, 0, 0, 44, 164, 306, 366, 292, 156, 54, 11, 1

Offset: 0

N. J. A. Sloane, Sep 05 2015

A242105 gives the first nonzero terms per row, without repetitions. - Reinhard Zumkeller, Sep 06 2015

			Triangle begins:
1,
1,1,
0,2,1,
0,2,3,1,
0,2,5,4,1,
0,0,7,9,5,1,
0,0,7,16,14,6,1,
0,0,7,23,30,20,7,1,
0,0,7,30,53,50,27,8,1,
0,0,7,37,83,103,77,35,9,1,
0,0,0,44,120,186,180,112,44,10,1,
0,0,0,44,164,306,366,292,156,54,11,1,
...

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
L. Haddad and C. Helou, Finite Sequences Dominated by the Squares, Journal of Integer Sequences, Volume 18, 2015, Issue 1, Article 15.1.8.

Cf. A242105, A261930 (row sums).

a261897 n k = a261897_tabl !! n !! (k-1)
a261897_row n = a261897_tabl !! n
a261897_tabl = [1] : f 1 0 [1] where
   f t h xs | t <= (h + 1) ^ 2  = ys : f (t + 1) h ys
            | otherwise         = ys' : f (t + 1) (h + 1) ys'
            where ys = zipWith (+) ([0] ++ xs) (xs ++ [0])
                  ys' = zipWith (+) ([0] ++ xs) (us ++ (0:vs) ++ [0])
                  (us, _:vs) = splitAt h xs
-- Reinhard Zumkeller, Sep 06 2015

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).

Original entry on oeis.org

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

Offset: 0

Thomas Scheuerle, Aug 04 2022

nonn

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.

			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}.

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

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

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)))

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).

cp's OEIS Frontend

A261897 Triangle read by rows: T(n,k) (1 <= k <= n+1) = number of sequences of length n, dominated by the squares, with entries from [0,k] and largest entry k.

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