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

Author

Roger L. Bagula, Feb 20 2009

Keywords

• nonn
• tabl
• easy

Comments

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

Examples

			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;
  ...
		

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

Crossrefs

A092956 (row sums for n > 0).

Cf. A105725.

Programs

Mathematica

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

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

Formula

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