|
| |
|
|
A162315
|
|
Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.
|
|
2
| |
|
|
1, 3, 1, 1, 6, 1, 3, 3, 9, 1, 1, 12, 6, 12, 1, 3, 5, 30, 10, 15, 1, 1, 18, 15, 60, 15, 18, 1, 3, 7, 63, 35, 105, 21, 21, 1, 1, 24, 28, 168, 70, 168, 28, 24, 1, 3, 9, 108, 84, 378, 126, 252, 36, 27, 1, 1, 30, 45, 360, 210, 756, 210, 360, 45, 30, 1
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Row reversed version of A124846. For the signless version of the inverse array and its connection with sums of powers of odd integers see A162313.
|
|
|
FORMULA
| TABLE ENTRIES
(1)... T(n,k) = (2 - (-1)^(n-k))*binomial(n,k).
GENERATING FUNCTION
(2)... exp(x*t)*(2*exp(t)-exp(-t)) = 1 + (3+x)*t + (1+6*x+x^2)*t^2/2!
+ ....
The e.g.f. can also be written as
(3)... exp(x*t)/G(-t), where G(t) = exp(t)/(2-exp(2*t)) is the e.g.f.
for A080253.
MISCELLANEOUS
The row polynomials form an Appell sequence of polynomials.
Row sums = A151821.
|
|
|
EXAMPLE
| Triangle begins
=================================================
n\k|..0.....1.....2.....3.....4.....5.....6.....7
=================================================
0..|..1
1..|..3.....1
2..|..1.....6.....1
3..|..3.....3.....9.....1
4..|..1....12.....6....12.....1
5..|..3.....5....30....10....15.....1
6..|..1....18....15....60....15....18.....1
7..|..3.....7....63....35...105....21....21.....1
...
|
|
|
MAPLE
| #A162315
T:=(n, k)->(2-(-1)^(n-k))*binomial(n, k):
for n from 0 to 10 do seq(T(n, k), k = 0..n) od;
|
|
|
CROSSREFS
| A007318, A151821 (row sums), A080253, A124846, A162313 (unsigned matrix inverse).
Sequence in context: A069972 A115017 A088439 * A109446 A088441 A061857
Adjacent sequences: A162312 A162313 A162314 * A162316 A162317 A162318
|
|
|
KEYWORD
| easy,nonn
|
|
|
AUTHOR
| Peter Bala (pbala(AT)talktalk.net), Jul 01 2009
|
|
|
EXTENSIONS
| Row sums corrected by Peter Bala (pbala(AT)talktalk.net), Apr 01 2010
|
| |
|
|
