A380841 - OEIS

%I #15 Feb 06 2025 10:23:26

%S 1,0,1,0,1,1,0,4,2,1,0,21,10,3,1,0,148,66,18,4,1,0,1305,560,141,28,5,

%T 1,0,13806,5770,1380,252,40,6,1,0,170401,69852,16095,2776,405,54,7,1,

%U 0,2403640,970886,217458,35940,4940,606,70,8,1,0,38143377,15228880,3335745,533304,70045,8088,861,88,9,1

%N Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

%F A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - _Seiichi Manyama_, Feb 06 2025

%e Array begins as:

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

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

%e   0,    4,   10,    18,    28,    40,     54, ...

%e   0,   21,   66,   141,   252,   405,    606, ...

%e   0,  148,  560,  1380,  2776,  4940,   8088, ...

%e   0, 1305, 5770, 16095, 35940, 70045, 124350, ...

%e   ...

%t A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

%Y Cf. A380843 (antidiagonal sums).

%Y Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.

%Y Rows n=0..2 give A000012, A001477, A028552.

%Y Main diagonal gives A380842.

%Y A(n,n+1) gives A213643(n+1).

%K nonn,tabl,new

%O 0,8

%A _Stefano Spezia_, Feb 05 2025