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 A381082 #17 Feb 26 2025 19:22:44
%S A381082 1,-2,1,4,-3,1,-8,8,-3,1,16,-18,11,-2,1,-32,44,-20,15,0,1,64,-80,94,5,
%T A381082 25,3,1,-128,272,56,294,105,49,7,1,256,112,1868,1596,1169,392,98,12,1,
%U A381082 -512,5280,12216,16148,10290,4305,1092,186,18,1
%N A381082 Triangle T(n,k) read by rows, where the columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=2.
%F A381082 T(n,k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), where m = 2.
%e A381082 Triangle starts:
%e A381082 [0] 1;
%e A381082 [1] -2, 1;
%e A381082 [2] 4, -3, 1;
%e A381082 [3] -8, 8, -3, 1;
%e A381082 [4] 16, -18, 11, -2, 1;
%e A381082 [5] -32, 44, -20, 15, 0, 1;
%e A381082 [6] 64, -80, 94, 5, 25, 3, 1;
%e A381082 [7] -128, 272, 56, 294, 105, 49, 7, 1;
%e A381082 [8] 256, 112, 1868, 1596, 1169, 392, 98, 12, 1;
%e A381082 [9] -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1;
%e A381082 ...
%p A381082 T:=(m,n,k)->add(Stirling1(n-i,k)*binomial(n,i)*m^(i)*(-1)^(n-k), i=0..n):
%p A381082 m:=2:seq(print(seq(T(m,n,k), k=0..n)), n=0..9);
%Y A381082 Cf. A000023 (row sums).
%Y A381082 Columns 0,1: A122803, A346397.
%Y A381082 Triangles: for m = -3 is A327997; for m = -2 is A137346 (unsigned); for m = -1 is A094816; for m = 0 is A132393; for m = 1 is A269953.
%K A381082 sign,tabl
%O A381082 0,2
%A A381082 _Igor Victorovich Statsenko_, Feb 13 2025