A135356 -id:A135356 - 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

A138569 First differences of A137776.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 1, 0, 4, -2, 10, -4, 20, -4, 36, 0, 64, 8, 120, 16, 240, 16, 496, 0, 1024, -32, 2080, -64, 4160, -64, 8256, 0, 16384, 128, 32640, 256, 65280, 256, 130816, 0, 262144, -512, 524800, -1024, 1049600, -1024, 2098176, 0, 4194304, 2048, 8386560, 4096, 16773120

Offset: 0

Paul Curtz, May 12 2008

sign

Index entries for linear recurrences with constant coefficients, signature (0, 4, 0, -6, 0, 4).

Cf. A135356.

a(n)=4a(n-2)-6a(n-4)+4a(n-6), n>7.
a(2n)=A000749(n). a(2n+1)=(-1)^(n+1)*A009116(n-1), n>0.
O.g.f.: x(1-x)(x^4-2x^2+1+x-2x^3+2x^5)/((1-2x^2)(1-2x^2+2x^4)). - R. J. Mathar, Aug 02 2008

Edited and extended by R. J. Mathar, Aug 02 2008

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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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