A346397 -id:A346397 - cp's OEIS frontend

A346398 Expansion of e.g.f. -log(1 - x) * exp(-3*x).

0, 1, -5, 20, -72, 249, -825, 2736, -8568, 29385, -74709, 417636, 698544, 21853233, 244181223, 3608612208, 54277152624, 878859416817, 15072037479099, 273539358115092, 5235734703888648, 105419854939796937, 2227408664800976487, 49278475088626210704, 1139260699549648412856

Offset: 0

Ilya Gutkovskiy, Jul 15 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A002741, A010843, A346397.

nmax = 24; CoefficientList[Series[-Log[1 - x] Exp[-3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-3)^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 24}]

a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i-4)*v[i]+3*(i-1)*v[i-1]+(-3)^(i-1)); v; \\ Seiichi Manyama, May 27 2022

a(n) = n! * Sum_{k=0..n-1} (-3)^k / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (n-4) * a(n-1) + 3 * (n-1) * a(n-2) + (-3)^(n-1). - Seiichi Manyama, May 27 2022
a(n) ~ exp(-3) * (n-1)!. - Vaclav Kotesovec, Jun 08 2022

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(-mx)/k! for the case m=2.

Original entry on oeis.org

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

Igor Victorovich Statsenko, Feb 13 2025

			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;
  ...

Cf. A000023 (row sums).
Columns 0,1: A122803, A346397.
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.

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);

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

cp's OEIS Frontend

A346398 Expansion of e.g.f. -log(1 - x) * exp(-3*x).

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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(-mx)/k! for the case m=2.

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cp's OEIS Frontend

A346398 Expansion of e.g.f. -log(1 - x) * exp(-3*x).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

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(-mx)/k! for the case m=2.