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

0,6

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

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)

Example

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;

Mathematica

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

Prog

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

Crossrefs

Cf. A048993, A029767, A180119.

Sequence in context: A291799 A295027 A225479 * A303439 A303345 A175802

Adjacent sequences: A156812 A156813 A156814 * A156816 A156817 A156818

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 16 2009

Extensions

Edited by G. C. Greubel, Jun 10 2021

Status

approved