A350365 -id:A350365 - cp's OEIS frontend

A350364 Array read by antidiagonals: T(n,k) is the number of sequences of length n with terms in 1..k such that all Hankel matrices of an odd number of consecutive terms are invertible, n, k >= 0.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 0, 0, 1, 4, 9, 6, 0, 0, 1, 5, 16, 24, 10, 0, 0, 1, 6, 25, 58, 66, 14, 0, 0, 1, 7, 36, 118, 212, 174, 20, 0, 0, 1, 8, 49, 208, 560, 758, 462, 20, 0, 0, 1, 9, 64, 334, 1206, 2620, 2722, 1178, 22, 0, 0

Offset: 0

Pontus von Brömssen, Dec 27 2021

T(n,2) = 0 for n >= 15.

For a fixed k, what can be said about T(n,k) as n grows? (For k <= 2, T(n,k) = 0 for large n.)

			Array begins:
  n\k|  0  1  2    3     4      5       6       7
  ---+-------------------------------------------
   0 |  1  1  1    1     1      1       1       1
   1 |  0  1  2    3     4      5       6       7
   2 |  0  1  4    9    16     25      36      49
   3 |  0  0  6   24    58    118     208     334
   4 |  0  0 10   66   212    560    1206    2282
   5 |  0  0 14  174   758   2620    6932   15506
   6 |  0  0 20  462  2722  12277   39871  105405
   7 |  0  0 20 1178  9628  57084  228451  714878
   8 |  0  0 22 3036 34132 265659 1309476 4849364

Pontus von Brömssen, Antidiagonals n = 0..14, flattened

Cf. A350330, A350365.

Cf. A000012 (row n = 0), A001477 (row n = 1), A000290 (row n = 2), A000007 (column k = 0), A130716 (column k = 1).

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.

Original entry on oeis.org

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

cp's OEIS Frontend

A350364 Array read by antidiagonals: T(n,k) is the number of sequences of length n with terms in 1..k such that all Hankel matrices of an odd number of consecutive terms are invertible, n, k >= 0.

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