A381082 Triangle T(m, n, k) for m = 2, 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!.

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, 25, 3, 1, -128, 272, 56, 294, 105, 49, 7, 1, 256, 112, 1868, 1596, 1169, 392, 98, 12, 1, -512, 5280, 12216, 16148, 10290, 4305, 1092, 186, 18, 1

Offset

0,2

Links

Table of n, a(n) for n=0..54.

Formula

T(m, n, k) = Sum_{i=0..n} Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), for m = 2.

Example

Triangle starts:
  [0]     1;
  [1]    -2,      1;
  [2]     4,     -3,       1;
  [3]    -8,      8,      -3,       1;
  [4]    16,    -18,      11,      -2,       1;
  [5]   -32,     44,     -20,      15,       0,        1;
  [6]    64,    -80,      94,       5,      25,        3,     1;
  [7]  -128,    272,      56,     294,     105,       49,     7,     1;
  [8]   256,    112,    1868,    1596,    1169,      392,    98,    12,    1;
  [9]  -512,   5280,   12216,   16148,   10290,     4305,  1092,   186,   18,     1;
  ...

Maple

T:=(m, n, k)->add(Stirling1(n-i, k)*binomial(n, i)*m^(i)*(-1)^(n-k), i=0..n):
m:=2:seq(print(seq(T(m, n, k), k=0..n)), n=0..9);

Crossrefs

Column 0 gives A122803.

Column 1 gives A346397.

Row sums give A000023.

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.

Sequence in context: A103316 A332389 A140069 * A105851 A106195 A247023

Adjacent sequences: A381067 A381068 A381081 * A381083 A381084 A381085

Keyword

sign,tabl,new

Author

Igor Victorovich Statsenko, Feb 13 2025

Status

approved