A156290 Triangle read by rows: alternating binomial coefficients with signs.

1, -4, 1, 15, -6, 1, -56, 28, -8, 1, 210, -120, 45, -10, 1, -792, 495, -220, 66, -12, 1, 3003, -2002, 1001, -364, 91, -14, 1, -11440, 8008, -4368, 1820, -560, 120, -16, 1, 43758, -31824, 18564, -8568, 3060, -816, 153, -18, 1, -167960, 125970, -77520

Offset: 1

Author

Hartmut F. W. Hoft, Feb 07 2009

Keywords

• easy
• sign
• tabl

Comments

Alternating binomial coefficients in the closed form expression for sequence A156289.

The Example lines below show the connection with Pascal's triangle A007318.

Examples

			R(2,1)=-4, R(3,3)=1, R(4,2)=28.
Here is Pascal's triangle with the entries in the present triangle preceded by a *:
......................1
.....................1, 1
...................1, 2,*1
.................1, 3, 3, 1
................1, 4, 6,*4,*1
..............1, 5, 10, 10, 5, 1
............1, 6, 15, 20,*15,*6,*1
..........1, 7, 21, 35, 35, 21, 7, 1
........1, 8, 28, 56, 70,*56,*28,*8,*1
...
		

References

• T. Myers and L. Shapiro, Some applications of the sequence 1, 5, 22, 93, 386, ... to Dyck paths and ordered trees, Congressus Numerant., 204 (2010), 93-104.

Crossrefs

Coefficient factor in elements of sequence A156289, the inverse of lower triangular matrix A156308.

Cf. A007318.

Programs

Mathematica

R[m_] := Flatten[Table[(-1)^(k + j) Binomial[2 k, k + j], {k, 1, m}, {j, 1, k}]]

Formula

R(k,j)=(-1)^(k+j)*Binomial(2k,k+j), for 1<= j<=k, and 0 otherwise.

Extensions

Edited by N. J. A. Sloane, Apr 05 2011