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

1,2

Comments

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

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

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.

Links

Table of n, a(n) for n=1..48.

Formula

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

Example

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
...

Mathematica

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

Crossrefs

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

Cf. A007318.

Sequence in context: A229468 A319039 A107873 * A080419 A095307 A159764

Adjacent sequences: A156287 A156288 A156289 * A156291 A156292 A156293

Keyword

easy,sign,tabl

Author

Hartmut F. W. Hoft, Feb 07 2009

Extensions

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

Status

approved