A156815 Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 28, 36, 24, 0, 24, 180, 300, 240, 120, 0, 120, 1488, 3240, 3120, 1800, 720, 0, 720, 15120, 43344, 50400, 33600, 15120, 5040, 0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320, 0, 40320, 2570400, 13068000, 22377600, 20018880, 11430720, 4656960, 1451520, 362880

Offset: 0

Author

Roger L. Bagula, Feb 16 2009

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
  1;
  0,    1;
  0,    1,      2;
  0,    2,      6,      6;
  0,    6,     28,     36,     24;
  0,   24,    180,    300,    240,    120;
  0,  120,   1488,   3240,   3120,   1800,    720;
  0,  720,  15120,  43344,  50400,  33600,  15120,   5040;
  0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320;
		

References

• Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 99.

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A048993, A029767, A180119.

Programs

Magma

[Factorial(n)*StirlingSecond(n,k)/Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 10 2021

Mathematica

T[n_, k_] = n!*StirlingS2[n, k]/Binomial[n, k];
Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten

Sage

flatten([[factorial(n)*stirling_number2(n,k)/binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

Formula

T(n, k) = n!*StirlingS2(n, k)/binomial(n, k).

From G. C. Greubel, Jun 10 2021: (Start)

T(n, 1) = T(n, n) = n!.

T(n, 2) = 2*A029767(n+1).

T(n, n-1) = A180119(n). (End)

Extensions

Edited by G. C. Greubel, Jun 10 2021