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%I #15 Nov 20 2019 09:33:46
%S 1,-1,0,0,-1,-1,1,1,0,-1,1,2,3,3,2,-2,-1,1,4,7,9,-9,-11,-12,-11,-7,0,
%T 9,-9,-18,-29,-41,-52,-59,-59,-50,50,41,23,-6,-47,-99,-158,-217,-267,
%U 267,317,358,381,375,328,229,71,-146,-413,413,680,997,1355,1736,2111,2439,2668,2739,2593,2180
%N Complementary Aitken's array: triangle of numbers {a(n,k), n >= 0, 0<=k<=n} read by rows, defined by a(0,0)=1, a(n,0)=-a(n-1,n-1), a(n,k)=a(n,k-1)+a(n-1,k-1).
%C a(n,0) of the triangle is equal to A000587(n), the Rao Uppuluri-Carpenter numbers or complementary Bell numbers.
%H Chai Wah Wu, <a href="/A247108/b247108.txt">Rows n=0..50 of triangle, flattened</a>
%H D. Wuilquin, <a href="/A000587/a000587_1.pdf">Letters to N. J. A. Sloane, August 1984</a>
%e Triangle begins:
%e 00: 1
%e 01: -1 0
%e 02: 0 -1 -1
%e 03: 1 1 0 -1
%e 04: 1 2 3 3 2
%e 05: -2 -1 1 4 7 9
%e 06: -9 -11 -12 -11 -7 0 9
%e 07: -9 -18 -29 -41 -52 -59 -59 -50
%e 08: 50 41 23 -6 -47 -99 -158 -217 -267
%e 09: 267 317 358 381 375 328 229 71 -146 -413
%t a[0, 0] = 1;
%t a[n_, 0] := -a[n - 1, n - 1];
%t a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n, k - 1] + a[n - 1, k - 1];
%t a[_, _] = 0;
%t Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 20 2019 *)
%o (Python)
%o # require Python 3.2 or higher
%o from itertools import accumulate
%o A247108_list = blist = [1]
%o for _ in range(10**2):
%o ....b = -blist[-1]
%o ....blist = list(accumulate([b]+blist))
%o ....A247108_list += blist
%o (Haskell)
%o a247108 n k = a247108_tabl !! n !! k
%o a247108_row n = a247108_tabl !! n
%o a247108_tabl = iterate (\row -> scanl (+) (- last row) row) [1]
%o -- _Reinhard Zumkeller_, Nov 22 2014
%Y Cf. A000587, A011971, A000110.
%K sign,tabl
%O 0,12
%A _Chai Wah Wu_, Nov 19 2014
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