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A269952
Triangle read by rows, T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(-j,-n)*S2(j,k), S2 the Stirling set numbers A048993, for n>=0 and 0<=k<=n.
2
1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 8, 19, 9, 1, 0, 16, 65, 55, 14, 1, 0, 32, 211, 285, 125, 20, 1, 0, 64, 665, 1351, 910, 245, 27, 1, 0, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 0, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1
OFFSET
0,5
FORMULA
T(n, k) = S2(n+1, k+1) - S2(n, k+1).
EXAMPLE
1,
0, 1,
0, 2, 1,
0, 4, 5, 1,
0, 8, 19, 9, 1,
0, 16, 65, 55, 14, 1,
0, 32, 211, 285, 125, 20, 1,
0, 64, 665, 1351, 910, 245, 27, 1.
MAPLE
A269952 := (n, k) -> Stirling2(n+1, k+1) - Stirling2(n, k+1):
seq(seq(A269952(n, k), k=0..n), n=0..9);
MATHEMATICA
Flatten[ Table[ Sum[(-1)^(n-j) Binomial[-j, -n] StirlingS2[j, k], {j, 0, n}], {n, 0, 9}, {k, 0, n}]]
CROSSREFS
Variant: A143494 (the main entry for this triangle).
A005493 (row sums), A074051 (alt. row sums), A000079 (col. 1), A001047 (col. 2),
A016269 (col. 3), A025211 (col. 4), A000096 (diag. n,n-1), A215862 (diag. n,n-2),
A049444, A136124, A143491 (matrix inverse).
Sequence in context: A100887 A073592 A164994 * A361954 A342500 A247268
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Apr 10 2016
STATUS
approved