A156991 - OEIS

%I #33 Mar 22 2022 09:52:23

%S 1,0,1,0,2,6,0,6,24,60,0,24,120,360,840,0,120,720,2520,6720,15120,0,

%T 720,5040,20160,60480,151200,332640,0,5040,40320,181440,604800,

%U 1663200,3991680,8648640,0,40320,362880,1814400,6652800,19958400,51891840,121080960,259459200

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

%C Apart from the left column of (essentially) zeros, the same as A105725. - _R. J. Mathar_, Mar 02 2009

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

%H G. C. Greubel, <a href="/A156991/b156991.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F T(n, k) = RisingFactorial(n, k). - _Peter Luschny_, Mar 22 2022

%e Triangle begins as:

%e   1;

%e   0,     1;

%e   0,     2,      6;

%e   0,     6,     24,      60;

%e   0,    24,    120,     360,     840;

%e   0,   120,    720,    2520,    6720,    15120;

%e   0,   720,   5040,   20160,   60480,   151200,   332640;

%e   0,  5040,  40320,  181440,  604800,  1663200,  3991680,   8648640;

%e   0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200;

%e   ...

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

%o (PARI) for(n=0,10, for(k=0,n, print1(n!*binomial(n+k-1,n), ", "))) \\ _G. C. Greubel_, Nov 19 2017

%o (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

%o (Sage)

%o for k in range(9):

%o     print([rising_factorial(n, k) for n in range(k+1)])

%o # _Peter Luschny_, Mar 22 2022

%Y A092956 (row sums for n > 0).

%Y Cf. A105725.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Feb 20 2009