%I A136334
%S A136334 1,0,2,0,0,4,1,0,0,8,0,4,0,0,16,0,0,12,0,0,32,1,0,0,32,0,0,64,0,6,0,0,
%T A136334 80,0,0,128,0,0,24,0,0,192,0,0,256,1,0,0,80,0,0,448,0,0,512,0,8,0,0,240,
%U A136334 0,0,1024,0,0,1024
%V A136334 1,0,2,0,0,4,-1,0,0,8,0,-4,0,0,16,0,0,-12,0,0,32,1,0,0,-32,0,0,64,0,6,
               0,0,-80,0,0,128,
%W A136334 0,0,24,0,0,-192,0,0,256,-1,0,0,80,0,0,-448,0,0,512,0,-8,0,0,240,0,0,-1024,
               0,0,1024
%N A136334 Triangular sequence from both a cubic expansion polynomial and a three 
               deep polynomial recursion: Expansion polynomial: f(x,t)=1/(1 - 2*x*t 
               + t^3); Recursion polynomials: p(x, n) = 2*x*p(x, n - 1) - p(x, n 
               - 3);.
%C A136334 Row sums are A000071 (Fibonacci numbers A000045(n)-1).
%C A136334 This sequence was a designed experiment in Umbral Calculus
%C A136334 using a Weierstrass like cubic polynomial for the expansion base.
%C A136334 I was testing the recursion form in the polynomial as:
%C A136334 f(x,t)=1/(1-p(x,1)*t +t^m): m the depth of the recursion.
%C A136334 as a recursion of the form:
%C A136334 p(x,n)=p(x,1)*p(x,n-1)-p(x,n-m).
%F A136334 f(x,t)=1/(1 - 2*x*t + t^3); f(x,t)=Sum[q(x,n)*t^n,{n,1,Infinity}]; p(x,
               0)=1;p(x,1)=2*x;p(x,2)=4*x^2; p(x, n) = 2*x*p(x, n - 1) - p(x, n 
               - 3);
%t A136334 (*expansion polynomial*) Clear[p, a] p[t_] = 1/(1 - 2*x*t + t^3) g = 
               Table[ ExpandAll[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], 
               {n, 0, 10}]; a = Table[ CoefficientList[SeriesCoefficient[ Series[p[t], 
               {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a] (* recursion polynomial*) 
               Clear[p] p[x, 0] = 1; p[x, 1] = 2x; p[x, 2] = 4x^2; p[x_, n_] := 
               p[x, n] = 2*x*p[x, n - 1] - p[x, n - 3]; Table[ExpandAll[p[x, n]], 
               {n, 0, Length[g] - 1}]; Flatten[Table[CoefficientList[p[x, n], x], 
               {n, 0, Length[g] - 1}]]
%Y A136334 Cf. A000071, A000045.
%Y A136334 Sequence in context: A140668 A071390 A061669 this_sequence A155039 A106235 
               A118965
%Y A136334 Adjacent sequences: A136331 A136332 A136333 this_sequence A136335 A136336 
               A136337
%K A136334 tabl,uned,sign
%O A136334 1,3
%A A136334 Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 12 2008