A369738 - OEIS

%I #27 Jan 31 2024 08:05:59

%S 1,1,0,1,-1,0,1,-2,1,0,1,-3,2,-1,0,1,-4,3,4,1,0,1,-5,4,21,-20,-1,0,1,

%T -6,5,56,-63,8,1,0,1,-7,6,115,-104,-423,184,-1,0,1,-8,7,204,-95,-2464,

%U 1899,-464,1,0,1,-9,8,329,36,-8245,1696,15201,-1648,-1,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(1 - (1+x)^k).

%F T(0,k) = 1; T(n,k) = -k * (n-1)! * Sum_{j=1..min(k,n)} binomial(k-1,j-1) * T(n-j,k)/(n-j)!.

%F T(n,k) = Sum_{j=0..n} k^j * Stirling1(n,j) * A000587(j).

%e Square array T(n,k) begins:

%e   1,  1,   1,    1,     1,      1,       1, ...

%e   0, -1,  -2,   -3,    -4,     -5,      -6, ...

%e   0,  1,   2,    3,     4,      5,       6, ...

%e   0, -1,   4,   21,    56,    115,     204, ...

%e   0,  1, -20,  -63,  -104,    -95,      36, ...

%e   0, -1,   8, -423, -2464,  -8245,  -21096, ...

%e   0,  1, 184, 1899,  1696, -21275, -124344, ...

%o (PARI) a000587(n) = sum(k=0, n, (-1)^k*stirling(n, k, 2));

%o T(n, k) = sum(j=0, n, k^j*stirling(n, j, 1)*a000587(j));

%Y Columns k=0..5 give A000007, A033999, (-1)^n * A062267(n), A369751, A369752, A369753.

%Y Main diagonal gives A369754.

%Y Cf. A000587, A294042.

%K sign,tabl

%O 0,8

%A _Seiichi Manyama_, Jan 30 2024