A176993 Triangle t(n,m) read by rows which contains in row n integer values of n! * binomial(n+m+1,m+1) / binomial(n-m-1,m+1) sorted along increasing m.

-1, 3, 6, -20, 12, 210, 40, 360, -3024, 180, 840, 55440, 1008, 3360, 60480, -1235520, 6720, 18144, 151200, 32432400, 51840, 120960, 665280, 19958400, -980179200, 453600, 950400, 3991680, 51891840, 33522128640, 4435200, 8553600, 29652480, 242161920, 10897286400, -1279935820800

Offset

0,2

Comments

These are (rational) solutions x to the equation binomial(n+m+1,m+1) - x*binomial(n-m-1,m+1) = 0, post-multiplied by n!.

Row sums are -1, 3, -14, 222, -2624, 56460, -1170672, 32608464, -959382720, 33579416160, -1268753731200,...

Links

Table of n, a(n) for n=0..35.

Example

-1; # n=0, m=0
3; # n=1, m=1
6, -20; # n=2, m=0,2
12, 210; # n=3, m=0,3
40, 360, -3024; # n=4,  m=0,1,4
180, 840, 55440; # n=5, m=0,1,5
1008, 3360, 60480, -1235520; # n=6, m=0,1,2,6
6720, 18144, 151200, 32432400; # n=7, m=0,1,2,7
51840, 120960, 665280, 19958400, -980179200; # n=8, m=0,1,2,3,8
453600, 950400, 3991680, 51891840, 33522128640; # n=9, m=0,1,2,3,9
4435200, 8553600, 29652480, 242161920, 10897286400, -1279935820800; # n=10, m=0,1,2,3,4,10

Mathematica

t[n_, m_] = Binomial[n + (m + 1), (m + 1)] - x*Binomial[n - (m + 1), (m + 1)];
(* if the solution exists it is made part of the array: if not, it is deleted*)
Table[Flatten[Table[If[Solve[t[n, m] == 0, x] == {}, {}, n!
        x /. Solve[t[n, m] == 0, x]], {m, 0, n}], 1], {n, 0, 10}];
Flatten[%]

Crossrefs

Sequence in context: A290784 A355605 A356912 * A359963 A276748 A339639

Adjacent sequences: A176990 A176991 A176992 * A176994 A176995 A176996

Keyword

sign,tabf

Author

Roger L. Bagula, Dec 08 2010

Status

approved