A291980 Triangle read by rows, T(n, k) = n!*[t^k] ([x^n] exp(x*t)/(1 - log(1+x))) for 0<=k<=n.

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 4, 8, 6, 4, 1, 14, 20, 20, 10, 5, 1, 38, 84, 60, 40, 15, 6, 1, 216, 266, 294, 140, 70, 21, 7, 1, 600, 1728, 1064, 784, 280, 112, 28, 8, 1, 6240, 5400, 7776, 3192, 1764, 504, 168, 36, 9, 1

Offset

0,5

Links

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

Formula

T(n, k) = binomial(n, n - k)*Sum_{j=0..n - k} j!*Stirling1(n - k, j). - Detlef Meya, May 12 2024

Example

Triangle starts:
[1]
[1,      1]
[1,      2,    1]
[2,      3,    3,   1]
[4,      8,    6,   4,   1]
[14,    20,   20,  10,   5,   1]
[38,    84,   60,  40,  15,   6,  1]
[216,  266,  294, 140,  70,  21,  7, 1]
[600, 1728, 1064, 784, 280, 112, 28, 8, 1]

Maple

T_row := proc(n) exp(x*t)/(1 - log(1+x)): series(%, x, n+1):
seq(n!*coeff(coeff(%, x, n), t, k), k=0..n) end:
seq(T_row(n), n=0..10);

Mathematica

T[n_, k_] := Binomial[n, n - k]*Sum[j!*StirlingS1[n - k, j], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* Detlef Meya, May 12 2024 *)

Crossrefs

Row sums: A291981.

Columns: A006252 (c=1), A108125 (c=2).

Diagonals: A000217 (d=3), A007290 (d=4), A033488 (d=5).

Cf. A291978.

Sequence in context: A191579 A097724 A091836 * A238281 A080850 A247453

Adjacent sequences: A291977 A291978 A291979 * A291981 A291982 A291983

Keyword

nonn,tabl

Author

Peter Luschny, Sep 15 2017

Status

approved