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A350529 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 1..k such that no iterated difference is zero, n, k >= 0.

%I A350529 #12 Jan 13 2022 02:32:31
%S A350529 1,1,0,1,1,0,1,2,0,0,1,3,2,0,0,1,4,6,2,0,0,1,5,12,10,2,0,0,1,6,20,32,
%T A350529 16,2,0,0,1,7,30,72,86,26,2,0,0,1,8,42,138,256,232,42,2,0,0,1,9,56,
%U A350529 234,624,906,622,68,2,0,0
%N A350529 Square array read by antidiagonals downwards: T(n,k) is the number of sequences of length n with terms in 1..k such that no iterated difference is zero, n, k >= 0.
%C A350529 For fixed n, T(n,k) is a quasi-polynomial of degree n in k. For example, T(4,k) = k^4 - (116/27)*k^3 + (25/3)*k^2 + b(k)*k + c(k), where b and c are periodic with period 6.
%H A350529 Pontus von Brömssen, <a href="/A350529/b350529.txt">Antidiagonals n = 0..20, flattened</a>
%e A350529   n\k|  0  1  2   3     4      5       6        7         8         9         10
%e A350529   ---+--------------------------------------------------------------------------
%e A350529    0 |  1  1  1   1     1      1       1        1         1         1          1
%e A350529    1 |  0  1  2   3     4      5       6        7         8         9         10
%e A350529    2 |  0  0  2   6    12     20      30       42        56        72         90
%e A350529    3 |  0  0  2  10    32     72     138      234       368       544        770
%e A350529    4 |  0  0  2  16    86    256     624     1278      2370      4030       6462
%e A350529    5 |  0  0  2  26   232    906    2790     6900     15096     29536      53678
%e A350529    6 |  0  0  2  42   622   3180   12366    36964     95494    215146     443464
%e A350529    7 |  0  0  2  68  1662  11116   54572   197294    601986   1562274    3652850
%e A350529    8 |  0  0  2 110  4426  38754  240278  1051298   3788268  11325490   30041458
%e A350529    9 |  0  0  2 178 11774 134902 1056546  5595236  23814458  82024662  246853482
%e A350529   10 |  0  0  2 288 31316 469306 4643300 29762654 149631992 593798912 2027577296
%e A350529 For n = 4 and k = 3, the following T(4,3) = 16 sequences are counted: 1212, 1213, 1312, 1313, 1323, 2121, 2131, 2132, 2312, 2313, 2323, 3121, 3131, 3132, 3231, 3232.
%o A350529 (Python)
%o A350529 def A350529_col(k,nmax):
%o A350529     d = []
%o A350529     c = [0]*(nmax+1)
%o A350529     while 1:
%o A350529         if not d or all(d[-1]):
%o A350529             c[len(d)] += 1 + (bool(d) and 2*d[0][0] != k+1)
%o A350529             if len(d) < nmax:
%o A350529                 d.append([0])
%o A350529                 for i in range(len(d)-1):
%o A350529                     d[-1].append(d[-1][-1]-d[-2][i])
%o A350529         while d and d[-1][0] == k:
%o A350529             d.pop()
%o A350529         if not d or len(d) == 1 and 2*d[0][0] >= k: return c
%o A350529         for i in range(len(d)):
%o A350529             d[-1][i] += 1
%Y A350529 Cf. A200154, A350364, A350530.
%Y A350529 Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A055232 (terms of row n=3 divided by 2).
%Y A350529 Columns: A000007 (k=0), A019590 (k=1), A040000 (k=2), A054886 (k=3).
%K A350529 nonn,tabl
%O A350529 0,8
%A A350529 _Pontus von Brömssen_, Jan 03 2022