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A247108 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).

%I A247108 #18 Jan 04 2025 15:15:26
%S A247108 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 A247108 9,-9,-18,-29,-41,-52,-59,-59,-50,50,41,23,-6,-47,-99,-158,-217,-267,
%U A247108 267,317,358,381,375,328,229,71,-146,-413,413,680,997,1355,1736,2111,2439,2668,2739,2593,2180
%N A247108 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 A247108 a(n,0) of the triangle is equal to A000587(n), the Rao Uppuluri-Carpenter numbers or complementary Bell numbers.
%H A247108 Chai Wah Wu, <a href="/A247108/b247108.txt">Rows n=0..50 of triangle, flattened</a>
%H A247108 D. Wuilquin, <a href="/A000587/a000587_1.pdf">Letters to N. J. A. Sloane, August 1984</a>
%e A247108 Triangle begins:
%e A247108 00:      1
%e A247108 01:     -1      0
%e A247108 02:      0     -1     -1
%e A247108 03:      1      1      0     -1
%e A247108 04:      1      2      3      3      2
%e A247108 05:     -2     -1      1      4      7      9
%e A247108 06:     -9    -11    -12    -11     -7      0      9
%e A247108 07:     -9    -18    -29    -41    -52    -59    -59    -50
%e A247108 08:     50     41     23     -6    -47    -99   -158   -217   -267
%e A247108 09:    267    317    358    381    375    328    229     71   -146   -413
%t A247108 a[0, 0] = 1;
%t A247108 a[n_, 0] := -a[n - 1, n - 1];
%t A247108 a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n, k - 1] + a[n - 1, k - 1];
%t A247108 a[_, _] = 0;
%t A247108 Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 20 2019 *)
%o A247108 (Python)
%o A247108 from itertools import accumulate
%o A247108 A247108_list = blist = [1]
%o A247108 for _ in range(10**2):
%o A247108     b = -blist[-1]
%o A247108     blist = list(accumulate([b]+blist))
%o A247108     A247108_list += blist
%o A247108 (Haskell)
%o A247108 a247108 n k = a247108_tabl !! n !! k
%o A247108 a247108_row n = a247108_tabl !! n
%o A247108 a247108_tabl = iterate (\row -> scanl (+) (- last row) row) [1]
%o A247108 -- _Reinhard Zumkeller_, Nov 22 2014
%Y A247108 Cf. A000587, A011971, A000110.
%K A247108 sign,tabl
%O A247108 0,12
%A A247108 _Chai Wah Wu_, Nov 19 2014