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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. 3
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, 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, 234, 624, 906, 622, 68, 2, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

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.

LINKS

Pontus von Brömssen, Antidiagonals n = 0..20, flattened

EXAMPLE

n\k| 0 1 2 3 4 5 6 7 8 9 10

---+--------------------------------------------------------------------------

0 | 1 1 1 1 1 1 1 1 1 1 1

1 | 0 1 2 3 4 5 6 7 8 9 10

2 | 0 0 2 6 12 20 30 42 56 72 90

3 | 0 0 2 10 32 72 138 234 368 544 770

4 | 0 0 2 16 86 256 624 1278 2370 4030 6462

5 | 0 0 2 26 232 906 2790 6900 15096 29536 53678

6 | 0 0 2 42 622 3180 12366 36964 95494 215146 443464

7 | 0 0 2 68 1662 11116 54572 197294 601986 1562274 3652850

8 | 0 0 2 110 4426 38754 240278 1051298 3788268 11325490 30041458

9 | 0 0 2 178 11774 134902 1056546 5595236 23814458 82024662 246853482

10 | 0 0 2 288 31316 469306 4643300 29762654 149631992 593798912 2027577296

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.

PROG

(Python)

def A350529_col(k, nmax):

d = []

c = [0]*(nmax+1)

while 1:

if not d or all(d[-1]):

c[len(d)] += 1 + (bool(d) and 2*d[0][0] != k+1)

if len(d) < nmax:

d.append([0])

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

CROSSREFS

Cf. A200154, A350364, A350530.

Rows: A000012 (n=0), A001477 (n=1), A002378 (n=2), A055232 (terms of row n=3 divided by 2).

Columns: A000007 (k=0), A019590 (k=1), A040000 (k=2), A054886 (k=3).

Sequence in context: A217315 A217593 A353434 * A322279 A350365 A331923

Adjacent sequences: A350526 A350527 A350528 * A350530 A350531 A350532

KEYWORD

nonn,tabl

AUTHOR

Pontus von Brömssen, Jan 03 2022

STATUS

approved

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Last modified January 29 00:02 EST 2023. Contains 359905 sequences. (Running on oeis4.)