A162315 Triangular array 2*P - P^-1, where P is Pascal's triangle A007318.

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

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.

Links

Table of n, a(n) for n=0..65.

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,tabl

Author

Peter Bala, Jul 01 2009

Extensions

Row sums corrected by Peter Bala, Apr 01 2010

Status

approved