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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).
1
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, 9, -9, -18, -29, -41, -52, -59, -59, -50, 50, 41, 23, -6, -47, -99, -158, -217, -267, 267, 317, 358, 381, 375, 328, 229, 71, -146, -413, 413, 680, 997, 1355, 1736, 2111, 2439, 2668, 2739, 2593, 2180
OFFSET
0,12
COMMENTS
a(n,0) of the triangle is equal to A000587(n), the Rao Uppuluri-Carpenter numbers or complementary Bell numbers.
EXAMPLE
Triangle begins:
00: 1
01: -1 0
02: 0 -1 -1
03: 1 1 0 -1
04: 1 2 3 3 2
05: -2 -1 1 4 7 9
06: -9 -11 -12 -11 -7 0 9
07: -9 -18 -29 -41 -52 -59 -59 -50
08: 50 41 23 -6 -47 -99 -158 -217 -267
09: 267 317 358 381 375 328 229 71 -146 -413
MATHEMATICA
a[0, 0] = 1;
a[n_, 0] := -a[n - 1, n - 1];
a[n_, k_] /; 0 <= k <= n := a[n, k] = a[n, k - 1] + a[n - 1, k - 1];
a[_, _] = 0;
Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 20 2019 *)
PROG
(Python)
# require Python 3.2 or higher
from itertools import accumulate
A247108_list = blist = [1]
for _ in range(10**2):
....b = -blist[-1]
....blist = list(accumulate([b]+blist))
....A247108_list += blist
(Haskell)
a247108 n k = a247108_tabl !! n !! k
a247108_row n = a247108_tabl !! n
a247108_tabl = iterate (\row -> scanl (+) (- last row) row) [1]
-- Reinhard Zumkeller, Nov 22 2014
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Chai Wah Wu, Nov 19 2014
STATUS
approved