This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380841 #15 Feb 06 2025 10:23:26
%S A380841 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 A380841 1,0,13806,5770,1380,252,40,6,1,0,170401,69852,16095,2776,405,54,7,1,
%U A380841 0,2403640,970886,217458,35940,4940,606,70,8,1,0,38143377,15228880,3335745,533304,70045,8088,861,88,9,1
%N A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.
%F A380841 A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - _Seiichi Manyama_, Feb 06 2025
%e A380841 Array begins as:
%e A380841 1, 1, 1, 1, 1, 1, 1, ...
%e A380841 0, 1, 2, 3, 4, 5, 6, ...
%e A380841 0, 4, 10, 18, 28, 40, 54, ...
%e A380841 0, 21, 66, 141, 252, 405, 606, ...
%e A380841 0, 148, 560, 1380, 2776, 4940, 8088, ...
%e A380841 0, 1305, 5770, 16095, 35940, 70045, 124350, ...
%e A380841 ...
%t A380841 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 A380841 Cf. A380843 (antidiagonal sums).
%Y A380841 Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.
%Y A380841 Rows n=0..2 give A000012, A001477, A028552.
%Y A380841 Main diagonal gives A380842.
%Y A380841 A(n,n+1) gives A213643(n+1).
%K A380841 nonn,tabl
%O A380841 0,8
%A A380841 _Stefano Spezia_, Feb 05 2025