cp's OEIS Frontend

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

%I A140414 #10 Jan 31 2025 05:48:24
%S A140414 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,
%T A140414 -7,3,8,-28,56,-70,56,-28,8,1,9,-36,84,-126,126,-84,36,-9,3,10,-45,
%U A140414 120,-210,252,-210,120,-45,10,1
%N A140414 Triangle T(p,s) showing the coefficients of sequences which are half their p-th differences.
%C A140414 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).
%C A140414 Setting this equal to 2*a(n) as demanded gives a recurrence with coefficients tabulated here,
%C A140414 a(n+p) = sum_{s=1..p} T(p,s)*a(n+p-s).
%F A140414 T(p,p) = 3 if p odd, =1 if p even. T(p,s) = (-1)^(s+1)*A014410(p,s), s<p.
%F A140414 Sum_{s=0..p} T(p,s) = 3.
%F A140414 Sum_{s=0..p} |T(p,s)| = A062510(n+1).
%e A140414 The triangle starts in row p=0 as:
%e A140414    3; (p=1, example A000244, a(n+1)=3*a(n) )
%e A140414    2,  1; (p=2 example A000244 or A000129, a(n+2) = 2*a(n+1)+a(n) )
%e A140414    3, -3,  3; (p=3 example A052103 or A136297, a(n+3) = 3*a(n+2)-3*a(n+1)+3*a(n) )
%e A140414    4, -6,  4,   1;
%e A140414    5,-10, 10,  -5,  3;
%e A140414    6,-15, 20, -15,  6,   1;
%e A140414    7,-21, 35, -35, 21,  -7,  3;
%e A140414    8,-28, 56, -70, 56, -28,  8,  1;
%e A140414    9,-36, 84,-126,126, -84, 36, -9, 3;
%e A140414   10,-45,120,-210,252,-210,120,-45,10,1;
%Y A140414 Cf. A135356.
%K A140414 sign,tabl,easy
%O A140414 1,1
%A A140414 _Paul Curtz_, Jun 25 2008
%E A140414 Edited by _R. J. Mathar_, Mar 02 2010