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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

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.

LINKS

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

EXAMPLE

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.

PROG

(Python)

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

CROSSREFS

Cf. A200154, A350365, A350529.

Rows: A000012 (n=1), A001477 (n=2), A007590 (n=3).

Columns: A000007 (k=0), A019590 (k=1), A130706 (k=2).

Sequence in context: A260019 A153036 A258651 * A258850 A182114 A122950

Adjacent sequences: A350527 A350528 A350529 * A350531 A350532 A350533

KEYWORD

nonn,tabl

AUTHOR

Pontus von Brömssen, Jan 03 2022

STATUS

approved

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Last modified February 6 05:07 EST 2023. Contains 360097 sequences. (Running on oeis4.)