login
A292894
Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).
4
1, 1, -1, 1, 0, 0, 1, 0, -2, 1, 1, 0, 0, -3, 1, 1, 0, 0, -6, 8, -2, 1, 0, 0, 0, -12, 55, -9, 1, 0, 0, 0, -24, -20, 84, -9, 1, 0, 0, 0, 0, -60, 330, -637, 50, 1, 0, 0, 0, 0, -120, -120, 2478, -4992, 267, 1, 0, 0, 0, 0, 0, -360, -210, 11704, -10593, 413, 1, 0, 0, 0, 0, 0, -720, -840, 19824, -15192, 92060, -2180
OFFSET
0,9
LINKS
FORMULA
From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (-1)^j * Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, ...
-1, 0, 0, 0, 0, ...
0, -2, 0, 0, 0, ...
1, -3, -6, 0, 0, ...
1, 8, -12, -24, 0, ...
-2, 55, -20, -60, -120, ...
PROG
(PARI) T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022
CROSSREFS
Columns k=0..2 give A000587, A292893, A292951.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Sequence in context: A212551 A125753 A185184 * A147701 A228348 A057516
KEYWORD
sign,tabl
AUTHOR
Seiichi Manyama, Sep 26 2017
STATUS
approved