A325873 T(n, k) = [x^k] Sum_{k=0..n} |Stirling1(n, k)|*FallingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 4, 0, 1, 0, 8, 5, 10, 0, 1, 0, 26, 58, 15, 20, 0, 1, 0, 194, 217, 238, 35, 35, 0, 1, 0, 1142, 2035, 1008, 728, 70, 56, 0, 1, 0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1

Offset

0,13

Links

Table of n, a(n) for n=0..54.

Example

Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 0,    1]
[3] [0, 1,    0,     1]
[4] [0, 1,    4,     0,     1]
[5] [0, 8,    5,     10,    0,    1]
[6] [0, 26,   58,    15,    20,   0,    1]
[7] [0, 194,  217,   238,   35,   35,   0,   1]
[8] [0, 1142, 2035,  1008,  728,  70,   56,  0,  1]
[9] [0, 9736, 13470, 11611, 3444, 1848, 126, 84, 0, 1]

Mathematica

p[n_] := Sum[Abs[StirlingS1[n, k]] FactorialPower[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten

Prog

(Sage)
def a_row(n):
    s = sum(stirling_number1(n, k)*falling_factorial(x, k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)]

Crossrefs

Cf. A079642 (variant), A129062, A325872.

Sequence in context: A271423 A019974 A344373 * A363973 A046781 A244530

Adjacent sequences: A325870 A325871 A325872 * A325874 A325875 A325876

Keyword

nonn,tabl

Author

Peter Luschny, Jun 27 2019

Status

approved