A136297 -id:A136297 - cp's OEIS frontend

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

Paul Curtz, Jun 25 2008

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).

			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;

Cf. A135356.

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

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

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

Edited by R. J. Mathar, Mar 02 2010

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

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