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A309386
Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(n,k) = Sum_{j=0..n} (-k)^(n-j)*Stirling2(n,j).
6
1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -1, 1, 1, 1, 1, -3, 1, 9, 2, 1, 1, 1, -4, 5, 19, -23, -9, 1, 1, 1, -5, 11, 25, -128, -25, 9, 1, 1, 1, -6, 19, 21, -343, 379, 583, 50, 1, 1, 1, -7, 29, 1, -674, 2133, 1549, -3087, -267, 1
OFFSET
0,18
LINKS
FORMULA
E.g.f. of column k: exp((1 - exp(-k*x))/k) for k > 0.
A(0,k) = 1 and A(n,k) = Sum_{j=0..n-1} (-k)^(n-1-j) * binomial(n-1,j) * A(j,k) for n > 0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 0, -1, -2, -3, -4, -5, ...
1, -1, -1, 1, 5, 11, 19, ...
1, 1, 9, 19, 25, 21, 1, ...
1, 2, -23, -128, -343, -674, -1103, ...
1, -9, -25, 379, 2133, 6551, 15211, ...
MATHEMATICA
T[n_, k_] := Sum[If[k == n-j == 0, 1, (-k)^(n-j)] * StirlingS2[n, j], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 07 2021 *)
CROSSREFS
Columns k=0..6 give A000012, (-1)^n * A000587(n), A009235, A317996, A318179, A318180, A318181.
Rows n=0+1, 2 give A000012, A024000.
Main diagonal gives A318183.
Sequence in context: A014649 A326568 A279817 * A253642 A070084 A352635
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Jul 27 2019
STATUS
approved