A113445 Triangle T(n,k) read by rows: consider the sequence a(m) = a(m-1) + sum_{0<j<=m/n} a(m-j*n) with a(0)=1. Row n of T(n,k) is formed by the coefficients of the recurrence relation of sequence b(i) = a(n*i).

1, 3, -1, 5, -6, 1, 7, -15, 11, -1, 9, -28, 38, -20, 1, 11, -45, 90, -90, 37, -1, 13, -66, 175, -260, 207, -70, 1, 15, -91, 301, -595, 707, -469, 135, -1, 17, -120, 476, -1176, 1862, -1848, 1052, -264, 1, 19, -153, 708, -2100, 4158, -5502, 4692, -2340, 521, -1

Offset

0,2

Comments

The sequence a(m) is also the expansion of (1-x^n)/(1-x-2x^n+x^{n+1}).

Instead of b(i) = a(n*i) one can take b(i) = a(n*i+p) for p=1..n-1.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

From Alois P. Heinz, Jul 16 2009: (Start)

T(n,k) = (-1)^k/(k+1)! * (1+k+(n-k)*2^(k+1)) * Product_{j=1..k}(n+j-k).

G.f. of column k: (-1)^k * x^k * (1+(2^(k+1)-1)*x)/(1-x)^(k+2). (End)

Example

For n=5 (A113444) the recurrence relation is b(i) = 11b(i-1)-45b(i-2) +90b(i-3)-90b(i-4)+37b(i-5)-b(i-6), so the fifth row reads 11, -45, 90, -90, 37, -1.

Maple

T:= (n, k)-> (-1)^k /(k+1)! *(1+k +(n-k) *2^(k+1)) *mul (n+j-k, j=1..k):
seq(seq(T(n, k), k=0..n), n=0..11);  # Alois P. Heinz, Jul 16 2009

Crossrefs

Columns k=0..2 give: A005408, -A000384, A007585(n-1) for n>=1. - Alois P. Heinz, Jul 16 2009

Sequence in context: A209758 A134083 A210551 * A108283 A208904 A344479

Adjacent sequences: A113442 A113443 A113444 * A113446 A113447 A113448

Keyword

sign,tabl

Author

Floor van Lamoen, Nov 04 2005

Extensions

More terms from Alois P. Heinz, Jul 16 2009

Status

approved