A142473 Triangle T(n, k) = n! * StirlingS1(n, k)/binomial(n, k), read by rows.

1, -1, 2, 4, -6, 6, -36, 44, -36, 24, 576, -600, 420, -240, 120, -14400, 13152, -8100, 4080, -1800, 720, 518400, -423360, 233856, -105840, 42000, -15120, 5040, -25401600, 18817920, -9455040, 3898944, -1411200, 463680, -141120, 40320, 1625702400, -1104606720, 510295680, -193777920, 64653120, -19595520, 5503680, -1451520, 362880

Offset

1,3

Comments

Row sums are: {1, 1, 4, -4, 276, -6348, 254976, -13188096, 887086080, ...}.

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

T(n, k) = n! * StirlingS1(n, k)/ binomial(n, k).

Example

The triangle begins as:
          1;
         -1,        2;
          4,       -6,        6;
        -36,       44,      -36,      24;
        576,     -600,      420,    -240,      120;
     -14400,    13152,    -8100,    4080,    -1800,    720;
     518400,  -423360,   233856, -105840,    42000, -15120,    5040;
  -25401600, 18817920, -9455040, 3898944, -1411200, 463680, -141120, 40320;

Maple

A142473:= (n, k)-> n!*Stirling1(n, k)/binomial(n, k);
seq(seq(A142473(n, k), k=1..n), n=1..12); # G. C. Greubel, Apr 02 2021

Mathematica

T[n_, k_]:= n!*StirlingS1[n, k]/Binomial[n, k];
Table[T[n, k], {n, 12}, {k, n}]//Flatten (* modified by G. C. Greubel, Apr 02 2021 *)

Prog

(Magma)
A142473:= func< n, k | Factorial(n)*StirlingFirst(n, k)/Binomial(n, k) >;
[A142473(n, k): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 02 2021
(Sage)
def A142473(n, k): return (-1)^(n-k)*factorial(n)*stirling_number1(n, k)/binomial(n, k)
flatten([[A142473(n, k) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Apr 02 2021

Crossrefs

Cf. A008275.

Sequence in context: A346911 A225187 A281485 * A132426 A072646 A091476

Adjacent sequences: A142470 A142471 A142472 * A142474 A142475 A142476

Keyword

sign,tabl

Author

Roger L. Bagula and Gary W. Adamson, Sep 21 2008

Extensions

Edited by G. C. Greubel, Apr 02 2021

Status

approved