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A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.

%I A326728 #14 Oct 30 2024 21:08:37
%S A326728 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 A326728 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 A326728 20,-9,0,-1,6,18,32,45,54,56,48,27,-10
%N A326728 A(n, k) = n*(k - 1)*k/2 - k, square array for n >= 0 and k >= 0 read by ascending antidiagonals.
%C A326728 A formal extension of the figurative numbers A139600 to negative n.
%H A326728 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 A326728 [0] 0, -1, -2, -3, -4, -5, -6,  -7,  -8,  -9, -10, ... A001489
%e A326728 [1] 0, -1, -1,  0,  2,  5,  9,  14,  20,  27,  35, ... A080956
%e A326728 [2] 0, -1,  0,  3,  8, 15, 24,  35,  48,  63,  80, ... A067998
%e A326728 [3] 0, -1,  1,  6, 14, 25, 39,  56,  76,  99, 125, ... A095794
%e A326728 [4] 0, -1,  2,  9, 20, 35, 54,  77, 104, 135, 170, ... A014107
%e A326728 [5] 0, -1,  3, 12, 26, 45, 69,  98, 132, 171, 215, ... A326725
%e A326728 [6] 0, -1,  4, 15, 32, 55, 84, 119, 160, 207, 260, ... A270710
%e A326728 [7] 0, -1,  5, 18, 38, 65, 99, 140, 188, 243, 305, ...
%p A326728 A := (n, k) -> n*(k - 1)*k/2 - k:
%p A326728 seq(seq(A(n - k, k), k=0..n), n=0..11);
%o A326728 (Python)
%o A326728 def A326728Row(n):
%o A326728     x, y = 1, 1
%o A326728     yield 0
%o A326728     while True:
%o A326728         yield -x
%o A326728         x, y = x + y - n, y - n
%o A326728 for n in range(8):
%o A326728     R = A326728Row(n)
%o A326728 print([next(R) for _ in range(11)])
%Y A326728 Cf. A001489 (n=0), A080956 (n=1), A067998 (n=2), A095794 (n=3), A014107 (n=4), A326725 (n=5), A270710 (n=6).
%Y A326728 Columns include A008585, A016933, A017329.
%Y A326728 Cf. A139600.
%K A326728 sign,tabl,easy
%O A326728 0,6
%A A326728 _Peter Luschny_, Aug 04 2019