A140414 Triangle T(p,s) showing the coefficients of sequences which are half their p-th differences.

3, 2, 1, 3, -3, 3, 4, -6, 4, 1, 5, -10, 10, -5, 3, 6, -15, 20, -15, 6, 1, 7, -21, 35, -35, 21, -7, 3, 8, -28, 56, -70, 56, -28, 8, 1, 9, -36, 84, -126, 126, -84, 36, -9, 3, 10, -45, 120, -210, 252, -210, 120, -45, 10, 1

Offset

1,1

Comments

The p-th differences of a sequence a(n) are Delta^p(n) = sum_{l=0}^p (-1)^(l+p)*binomial(p,l)*a(n+l).

Setting this equal to 2*a(n) as demanded gives a recurrence with coefficients tabulated here,

a(n+p) = sum_{s=1..p} T(p,s)*a(n+p-s).

Links

Table of n, a(n) for n=1..55.

Formula

T(p,p) = 3 if p odd, =1 if p even. T(p,s) = (-1)^(s+1)*A014410(p,s), s<p.

sum_{s=0..p} T(p,s) = 3.

sum_{s=0..p} |T(p,s)| = A062510(n+1).

Example

The triangle starts in row p=0 as:
.3; (p=1, example A000244, a(n+1)=3*a(n)
.2,..1; (p=2 example A000244 or A000129, a(n+2) = 2*a(n+1)+a(n) )
.3,.-3,..3; (p=3 example A052103 or A136297, a(n+3) = 3*a(n+2)-3*a(n+1)+3*a(n) )
.4,.-6,..4,...1;
.5,-10,.10,..-5,..3;
.6,-15,.20,.-15,..6,...1;
.7,-21,.35,.-35,.21,..-7,..3;
.8,-28,.56,.-70,.56,.-28,..8,..1;
.9,-36,.84,-126,126,.-84,.36,.-9,.3;
10,-45,120,-210,252,-210,120,-45,10,1;

Crossrefs

Cf. A135356.

Sequence in context: A244758 A230493 A128262 * A129514 A175506 A211705

Adjacent sequences: A140411 A140412 A140413 * A140415 A140416 A140417

Keyword

sign,tabl,easy

Author

Paul Curtz, Jun 25 2008

Extensions

Edited by R. J. Mathar, Mar 02 2010)

Status

approved