A094485 T(n, k) = Stirling1(n+1, k) - Stirling1(n, k-1), for 1 <= k <= n. Triangle read by rows.

-1, 2, -2, -6, 9, -3, 24, -44, 24, -4, -120, 250, -175, 50, -5, 720, -1644, 1350, -510, 90, -6, -5040, 12348, -11368, 5145, -1225, 147, -7, 40320, -104544, 105056, -54152, 15680, -2576, 224, -8, -362880, 986256, -1063116, 605556, -202041, 40824, -4914, 324, -9, 3628800, -10265760, 11727000, -7236800

Offset

1,2

Links

Table of n, a(n) for n=1..49.

Formula

E.g.f.: -x*y*(1+y)^(x-1). [T(n,k) = n!*[x^k]([y^n] -x*y*(y+1)^(x-1)).]

The matrix inverse of the Worpitzky triangle. More precisely:

T(n, k) = -n!*InvW(n, k) where InvW is the matrix inverse of A028246. - Peter Luschny, May 26 2020

Example

Triangle starts:
[n\k    1        2       3      4      5      6     7  8]
[1]    -1;
[2]     2,      -2;
[3]    -6,       9,     -3;
[4]    24,     -44,     24,     -4;
[5]  -120,     250,   -175,     50,    -5;
[6]   720,   -1644,   1350,   -510,    90,    -6;
[7] -5040,   12348, -11368,   5145, -1225,   147,   -7;
[8] 40320, -104544, 105056, -54152, 15680, -2576,  224,  -8;

Maple

T := (n, k) -> Stirling1(n+1, k) - Stirling1(n, k-1);
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, May 26 2020

Mathematica

Table[StirlingS1[n+1, k]-StirlingS1[n, k-1], {n, 10}, {k, n}]//Flatten (* Harvey P. Dale, Jul 25 2024 *)

Crossrefs

Cf. A019538, A028246, A163626.

Cf. A000142, A052881.

Sequence in context: A131553 A277510 A169800 * A331988 A242978 A345308

Adjacent sequences: A094482 A094483 A094484 * A094486 A094487 A094488

Keyword

easy,sign,tabl

Author

Vladeta Jovovic, Jun 05 2004

Extensions

Offset of k shifted and edited by Peter Luschny, May 26 2020

Status

approved