A350529 -id:A350529 - cp's OEIS frontend

A350530 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 0..k such that the (n-1)-st difference is zero, but no earlier iterated difference is zero, n, k >= 1.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 0, 0, 0, 1, 4, 2, 0, 0, 0, 1, 5, 4, 0, 0, 0, 0, 1, 6, 8, 0, 0, 0, 0, 0, 1, 7, 12, 4, 0, 0, 0, 0, 0, 1, 8, 18, 12, 8, 4, 0, 0, 0, 0, 1, 9, 24, 28, 36, 28, 4, 0, 0, 0, 0, 1, 10, 32, 52, 84, 116, 48, 16, 0, 0, 0, 0

Offset: 1

Pontus von Brömssen, Jan 03 2022

For fixed n, T(n,k) is a quasi-polynomial of degree n-1 in k. For example, T(4,k) = (8/27)*k^3 - 2*k^2 + b(k)*k + c(k), where b and c are periodic with period 3.

			Array begins:
  n\k|  0  1  2  3  4   5    6     7     8      9     10
  ---+--------------------------------------------------
   1 |  1  1  1  1  1   1    1     1     1      1      1
   2 |  0  1  2  3  4   5    6     7     8      9     10
   3 |  0  0  0  2  4   8   12    18    24     32     40
   4 |  0  0  0  0  0   4   12    28    52     84    132
   5 |  0  0  0  0  0   8   36    84   176    332    568
   6 |  0  0  0  0  4  28  116   308   704   1396   2548
   7 |  0  0  0  0  4  48  232   728  2104   4940  11008
   8 |  0  0  0  0 16 100  556  1936  7092  19908  49364
   9 |  0  0  0  0 12 176 1348  6588 23356  74228 202504
  10 |  0  0  0  0  8 268 2492 15544 72820 259800 842688
For n = 4 and k = 6, the following T(4,6) = 12 sequences are counted: 1454, 1564, 2125, 2565, 3126, 3236, 4541, 4651, 5212, 5652, 6213, 6323.

Pontus von Brömssen, Antidiagonals n = 1..19, flattened

Cf. A200154, A350365, A350529.
Rows: A000012 (n=1), A001477 (n=2), A007590 (n=3).
Columns: A000007 (k=0), A019590 (k=1), A130706 (k=2).

def A350530_col(k,nmax):
    d = []
    c = [0]*nmax
    while 1:
        if not d or all(d[-1][:-1]):
            if d and d[-1][-1] == 0:
                c[len(d)-1] += 1 + (0 != 2*d[0][0] != k+1)
            elif len(d) < nmax:
                d.append([-1])
                for i in range(len(d)-1):
                    d[-1].append(d[-1][-1]-d[-2][i])
        while d and d[-1][0] == k:
            d.pop()
        if not d or len(d) == 1 and 2*d[0][0] >= k: return c
        for i in range(len(d)):
            d[-1][i] += 1

A353434 Array read by descending antidiagonals: T(n,m) is the number of sequences of length n >= 0 with elements in 1..m-1 such that no iterated difference is divisible by m >= 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 4, 6, 2, 0, 0, 1, 5, 12, 8, 2, 0, 0, 1, 6, 20, 28, 6, 2, 0, 0, 1, 7, 30, 64, 48, 6, 2, 0, 0, 1, 8, 42, 126, 164, 60, 6, 2, 0, 0, 1, 9, 56, 216, 444, 336, 60, 6, 2, 0, 0, 1, 10, 72, 344, 954, 1350, 552, 52, 6, 2, 0, 0

Offset: 0

Pontus von Brömssen, Apr 21 2022

			Array begins:
  n\m| 1  2  3  4  5    6     7      8       9       10
  ---+-------------------------------------------------
   0 | 1  1  1  1  1    1     1      1       1        1
   1 | 0  1  2  3  4    5     6      7       8        9
   2 | 0  0  2  6 12   20    30     42      56       72
   3 | 0  0  2  8 28   64   126    216     344      512
   4 | 0  0  2  6 48  164   444    954    1850     3240
   5 | 0  0  2  6 60  336  1350   3630    8732    18240
   6 | 0  0  2  6 60  552  3582  11898   36290    90624
   7 | 0  0  2  6 52  772  8550  33862  133628   398048
   8 | 0  0  2  6 48 1054 17364  83946  437666  1545468
   9 | 0  0  2  6 48 1614 30126 182134 1278314  5300824
  10 | 0  0  2  6 48 2740 44922 346638 3321680 16079024

Pontus von Brömssen, Plot of T(n,7) for 0 <= n <= 200

Cf. A350529, A353433, A353436.
Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A245996 (n=3).
Columns: A000007 (m=1), A019590 (m=2), A040000 (m=3).

T(n,m) = A353433(n,m) if m is prime.
T(n,1) = 0 for n >= 1.
T(n,2) = 0 for n >= 2.
T(n,3) = 2 for n >= 1.
T(n,4) = 6 for n >= 4.
T(n,5) = 48 for n >= 8.
It appears that T(n,7) = T(n+42,7) for n >= 56. (See linked plot.)

cp's OEIS Frontend

A350530 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 0..k such that the (n-1)-st difference is zero, but no earlier iterated difference is zero, n, k >= 1.

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