A291977 Triangle read by rows, T(n, k) = Sum_{j=0..n} (-1)^(k-j)*Eulerian1(n, j)* binomial(n-j, n-k) for 0 <= k <= n.

1, 1, -1, 1, -1, 0, 1, 1, -4, 2, 1, 7, -16, 8, 0, 1, 21, -28, -26, 48, -16, 1, 51, 32, -356, 408, -136, 0, 1, 113, 492, -1774, 1072, 912, -1088, 272, 1, 239, 2592, -5008, -6656, 20736, -15872, 3968, 0, 1, 493, 10628, -50, -94432, 154528, -57856, -45056, 39680, -7936

Offset

0,9

Links

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

Formula

T(n, k) = Sum_{j=0..n} (-1)^(k-j)*A173018(n, j)*A007318(n-j, n-k) for 0 <= k <= n.

Example

Triangle starts:
0| 1
1| 1,  -1
2| 1,  -1,     0
3| 1,   1,    -4,    2
4| 1,   7,   -16,    8,       0
5| 1,  21,   -28,   -26,     48,    -16
6| 1,  51,    32,  -356,    408,   -136,      0
7| 1, 113,   492, -1774,   1072,    912,  -1088,    272
8| 1, 239,  2592, -5008,  -6656,  20736, -15872,   3968,     0
9| 1, 493, 10628,   -50, -94432, 154528, -57856, -45056, 39680, -7936
---------------------------------------------------------------------
k| 0    1      2      3       4       5       6       7      8      9

Maple

with(combinat):
T := (n, k) -> add((-1)^(k-j)*eulerian1(n, j)*binomial(n-j, n-k), j=0..n):
seq(print(seq(T(n, k), k=0..n)), n=0..9);

Prog

(Python)
from sympy.core.cache import cacheit
from sympy import binomial
@cacheit
def eulerian1(n, k): return 1 if k==0 else 0 if k==n else eulerian1(n - 1, k)*(k + 1) + eulerian1(n - 1, k - 1)*(n - k)
def T(n, k): return sum([(-1)**(k - j)*eulerian1(n, j)*binomial(n - j, n - k) for j in range(n + 1)])
for n in range(10): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Sep 11 2017

Crossrefs

Cf. A007318, A142073, A173018.

Sequence in context: A046741 A136249 A142147 * A142073 A193559 A135294

Adjacent sequences: A291974 A291975 A291976 * A291978 A291979 A291980

Keyword

sign,tabl

Author

Peter Luschny, Sep 10 2017

Status

approved