A155999 Triangle T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k), read by rows.

1, 0, 0, 0, 1, 0, 0, -3, -3, 0, 0, 6, 49, 6, 0, 0, -10, -375, -375, -10, 0, 0, 15, 2015, 8100, 2015, 15, 0, 0, -21, -8820, -105350, -105350, -8820, -21, 0, 0, 28, 33782, 1014300, 2893401, 1014300, 33782, 28, 0, 0, -36, -117810, -8004150, -54009270, -54009270, -8004150, -117810, -36, 0

Offset

0,8

Comments

Row sums are: {1, 0, 1, -6, 61, -770, 12160, -228382, 4989621, -124262532, 3475892685, ...}.

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Formula

T(n, k) = (-1)^n*StirlingS2(n, k)*StirlingS2(n, n-k)

Example

Triangle begins as:
  1;
  0,   0;
  0,   1,       0;
  0,  -3,      -3,        0;
  0,   6,      49,        6,         0;
  0, -10,    -375,     -375,       -10,         0;
  0,  15,    2015,     8100,      2015,        15,        0;
  0, -21,   -8820,  -105350,   -105350,     -8820,      -21,       0;
  0,  28,   33782,  1014300,   2893401,   1014300,    33782,      28,   0;
  0, -36, -117810, -8004150, -54009270, -54009270, -8004150, -117810, -36, 0;

Mathematica

T[n_, k_]:= (-1)^n*StirlingS2[n, k]*StirlingS2[n, n-k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten

Prog

(Sage)
def A155999(n, k): return (-1)^n*stirling_number2(n, k)*stirling_number2(n, n-k)
flatten([[A155999(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 27 2021
(Magma)
A155999:= func< n, k | (-1)^n*StirlingSecond(n, k)*StirlingSecond(n, n-k) >;
[A155999(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 27 2021
(PARI) T(n, k) = (-1)^n*stirling(n, k, 2)*stirling(n, n-k, 2);
matrix(10, 10, n, k, n--; k--; if (n>=k, T(n, k))) \\ Michel Marcus, Feb 27 2021

Crossrefs

Cf. A048993.

Sequence in context: A300175 A129533 A360849 * A338034 A299904 A221768

Adjacent sequences: A155996 A155997 A155998 * A156000 A156001 A156002

Keyword

tabl,sign

Author

Roger L. Bagula, Feb 01 2009

Extensions

Edited by G. C. Greubel, Feb 27 2021

Status

approved