A156991 Triangle T(n,k) read by rows: T(n,k) = n! * binomial(n + k - 1, n).

1, 0, 1, 0, 2, 6, 0, 6, 24, 60, 0, 24, 120, 360, 840, 0, 120, 720, 2520, 6720, 15120, 0, 720, 5040, 20160, 60480, 151200, 332640, 0, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200

Offset

0,5

Comments

Apart from the left column of (essentially) zeros, the same as A105725. - R. J. Mathar, Mar 02 2009

References

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Formula

T(n, k) = RisingFactorial(n, k). - Peter Luschny, Mar 22 2022

Example

Triangle begins as:
  1;
  0,     1;
  0,     2,      6;
  0,     6,     24,      60;
  0,    24,    120,     360,     840;
  0,   120,    720,    2520,    6720,    15120;
  0,   720,   5040,   20160,   60480,   151200,   332640;
  0,  5040,  40320,  181440,  604800,  1663200,  3991680,   8648640;
  0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200;
  ...

Mathematica

Table[n!*Binomial[n+k-1, n], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(PARI) for(n=0, 10, for(k=0, n, print1(n!*binomial(n+k-1, n), ", "))) \\ G. C. Greubel, Nov 19 2017
(Sage) flatten([[factorial(n)*binomial(n+k-1, n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 10 2021
(Sage)
for k in range(9):
    print([rising_factorial(n, k) for n in range(k+1)])
# Peter Luschny, Mar 22 2022

Crossrefs

A092956 (row sums for n > 0).

Cf. A105725.

Sequence in context: A350256 A345208 A241810 * A229586 A294789 A197035

Adjacent sequences: A156988 A156989 A156990 * A156992 A156993 A156994

Keyword

nonn,tabl,easy

Author

Roger L. Bagula, Feb 20 2009

Status

approved