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

Author

Peter Bala, Jul 01 2009

Keywords

• easy
• nonn
• tabl

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.

Examples

			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
...
		

Crossrefs

A007318, A151821 (row sums), A080253, A124846, A162313 (unsigned matrix inverse).

Programs

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;

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.

Extensions

Row sums corrected by Peter Bala, Apr 01 2010