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A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.
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%I #14 Oct 30 2024 21:08:37

%S 0,0,-1,0,-1,-2,0,-1,-1,-3,0,-1,0,0,-4,0,-1,1,3,2,-5,0,-1,2,6,8,5,-6,

%T 0,-1,3,9,14,15,9,-7,0,-1,4,12,20,25,24,14,-8,0,-1,5,15,26,35,39,35,

%U 20,-9,0,-1,6,18,32,45,54,56,48,27,-10

%N A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

%C A formal extension of the figurative numbers A139600 to negative n.

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/FigurateNumber">Figurate number — a very short introduction</a>. With plots from Stefan Friedrich Birkner.

%e [0] 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, ... A001489

%e [1] 0, -1, -1, 0, 2, 5, 9, 14, 20, 27, 35, ... A080956

%e [2] 0, -1, 0, 3, 8, 15, 24, 35, 48, 63, 80, ... A067998

%e [3] 0, -1, 1, 6, 14, 25, 39, 56, 76, 99, 125, ... A095794

%e [4] 0, -1, 2, 9, 20, 35, 54, 77, 104, 135, 170, ... A014107

%e [5] 0, -1, 3, 12, 26, 45, 69, 98, 132, 171, 215, ... A326725

%e [6] 0, -1, 4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710

%e [7] 0, -1, 5, 18, 38, 65, 99, 140, 188, 243, 305, ...

%p A := (n, k) -> n*(k - 1)*k/2 - k:

%p seq(seq(A(n - k, k), k=0..n), n=0..11);

%o (Python)

%o def A326728Row(n):

%o x, y = 1, 1

%o yield 0

%o while True:

%o yield -x

%o x, y = x + y - n, y - n

%o for n in range(8):

%o R = A326728Row(n)

%o print([next(R) for _ in range(11)])

%Y Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).

%Y Columns include A008585, A016933, A017329.

%Y Cf. A139600.

%K sign,tabl,easy,changed

%O 0,6

%A _Peter Luschny_, Aug 04 2019