A391008 Triangle read by rows: T(n, k) = [x^k] Sum_{j=0..n} FallingFactorial(n, j)*RisingFactorial(x, j).

1, 1, 1, 1, 4, 2, 1, 21, 24, 6, 1, 208, 348, 168, 24, 1, 3745, 7520, 4980, 1320, 120, 1, 106116, 237630, 189480, 68760, 11520, 720, 1, 4299589, 10407432, 9412410, 4158840, 960120, 110880, 5040, 1, 234834496, 603905624, 599743536, 304343760, 86392320, 13849920, 1169280, 40320

Offset

0,5

Links

Table of n, a(n) for n=0..44.

Example

Triangle starts:
  [0] [1]
  [1] [1,      1]
  [2] [1,      4,      2]
  [3] [1,     21,     24,      6]
  [4] [1,    208,    348,    168,    24]
  [5] [1,   3745,   7520,   4980,  1320,   120]
  [6] [1, 106116, 237630, 189480, 68760, 11520, 720]

Prog

(SageMath)
def Trow(n):
    s = sum(falling_factorial(n, k)*rising_factorial(x, k) for k in (0..n))
    return expand(s).list()
[Trow(n) for n in (0..9)]

Crossrefs

Cf. A008279 (falling factorial), A124320 (rising factorial), A046662 (row sums).

Sequence in context: A152391 A144088 A039948 * A111536 A111559 A224798

Adjacent sequences: A391005 A391006 A391007 * A391009 A391010 A391011

Keyword

nonn,tabl

Author

Peter Luschny, Dec 22 2025

Status

approved