A089515 Triangle of signed numbers used for the computation of the column sequences of triangle A090215.

1, -1, 5, 1, -35, 90, -3, 595, -6885, 12005, 143, -150535, 6175845, -39484445, 52245760, -58201, 316465625, -42458934375, 772604284375, -3322503800000, 3547818864576, 216931, -6012846875, 2544269990625, -120371747505625, 1294115230100000, -4145626343257056, 3713894747640000

Offset

1,3

Comments

A090215(n+m,m)= sum(a(m,p)*((p+3)*(p+2)*(p+1)*p)^n,p=1..m)/D(m) with D(m) := A089516(m); m=1,2,..., n>=0.

Links

Table of n, a(n) for n=1..28.

Wolfdieter Lang, First 7 rows.

Formula

a(n, m)= D(n)*((-1)^(n-m))*(fallfac(m+3, 4)^(n-1))/(product(fallfac(m+3, 4)-fallfac(r+3, 4), r=1..m-1)*product(fallfac(r+3, 4)-fallfac(m+3, 4), r=m+1..n)), with D(n) := A089516(n) and fallfac(n, m) := A008279(n, m) (falling factorials), 1<=m<=n else 0. (Replace in the denominator the first product by 1 if m=1 and the second one by 1 if m=n.)

Example

Triangle begins:
   1;
  -1,   5;
   1, -35,   90;
  -3, 595,-6885, 12005;
  ...
A090215(2+3,3) = 199296 = (1*(4*3*2*1)^2 - 35*(5*4*3*2)^2 + 90*(6*5*4*3)^2)/56.
a(3,2)= -35 = 56*(-1)*((5*4*3*2)^2)/((5*4*3*2-4*3*2*1)*(6*5*4*3-5*4*3*2)).

Crossrefs

Sequence in context: A334887 A039813 A160632 * A158820 A082437 A308440

Adjacent sequences: A089512 A089513 A089514 * A089516 A089517 A089518

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Dec 01 2003

Status

approved