A359759 - OEIS

%I #13 Jan 28 2023 12:17:09

%S 1,0,1,0,-3,1,0,13,-9,1,0,-103,79,-18,1,0,1241,-905,265,-30,1,0,

%T -19691,13771,-4290,665,-45,1,0,384805,-262885,82621,-14630,1400,-63,

%U 1,0,-8918351,6007247,-1888362,353381,-40390,2618,-84,1

%N Table read by rows. T(n, k) = (-1)^(n - k) * Sum_{j=k..n} binomial(n, j) * A354794(j, k) * j^(n - j).

%C Inspired by a formula of _Mélika Tebni_ in A048993.

%F E.g.f. of column k: (exp(LambertW(x*exp(-x))) - 1)^k / k!. (Note that (exp(-LambertW(-x*exp(-x))) - 1)^k / k! is the e.g.f. of column k of Stirling2.) - _Mélika Tebni_, Jan 27 2023

%e Triangle T(n, k) starts:

%e [0] 1;

%e [1] 0,         1;

%e [2] 0,        -3,          1;

%e [3] 0,        13,         -9,        1;

%e [4] 0,      -103,         79,      -18,        1;

%e [5] 0,      1241,       -905,      265,      -30,       1;

%e [6] 0,    -19691,      13771,    -4290,      665,     -45,      1;

%e [7] 0,    384805,    -262885,    82621,   -14630,    1400,    -63,    1;

%e [8] 0,  -8918351,    6007247, -1888362,   353381,  -40390,   2618,  -84,    1;

%e [9] 0, 238966705, -159432369, 50110705, -9627702, 1206471, -96138, 4494, -108, 1;

%p T := (n, k) -> (-1)^(n - k)*add(binomial(n, j) * A354794(j, k) * j^(n - j), j = k..n): for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

%Y Cf. A059297, A354794, A357247, A048993.

%K sign,tabl

%O 0,5

%A _Peter Luschny_, Jan 27 2023