A358366 - OEIS

A358366

Table read by rows. T(n, k) = [x^k] n! * Sum_{j=0..n} binomial(n*x, j).

%I #8 Nov 13 2022 10:36:23

%S 1,1,1,2,2,4,6,15,0,27,24,56,176,-128,256,120,470,125,3125,-3125,3125,

%T 720,2664,10944,-16200,71280,-69984,46656,5040,26796,17836,376957,

%U -840350,1882384,-1647086,823543,40320,204672,1022720,-2222080,16257024,-34865152,55050240,-41943040,16777216

%N Table read by rows. T(n, k) = [x^k] n! * Sum_{j=0..n} binomial(n*x, j).

%e [n\k] 0 1 2 3 4 5 6 7 [Sum]

%e --------------------------------------------------------------------------

%e [0] 1; [1]

%e [1] 1, 1; [2]

%e [2] 2, 2, 4; [8]

%e [3] 6, 15, 0, 27; [48]

%e [4] 24, 56, 176, -128, 256; [384]

%e [5] 120, 470, 125, 3125, -3125, 3125; [3840]

%e [6] 720, 2664, 10944, -16200, 71280, -69984, 46656; [46080]

%e [7] 5040, 26796, 17836, 376957, -840350, 1882384, -1647086, 823543; [645120]

%p Trow := proc(n) expand(n!*add(binomial(n*x, j), j = 0..n));

%p seq(coeff(%, x, k), k=0..n) end: seq(print(Trow(n)), n = 0..8);

%Y Cf. T(n, 0) = n! (A000142), T(n, n) = n^n (A000312), A000165 (row sums).

%K sign,tabl

%O 0,4

%A _Peter Luschny_, Nov 12 2022