A137526 - OEIS

%I #3 Mar 30 2012 17:34:26

%S 1,0,1,-2,0,1,0,-6,0,1,0,0,-12,0,1,0,0,0,-20,0,1,0,0,0,0,-30,0,1,0,0,

%T 0,0,0,-42,0,1,0,0,0,0,0,0,-56,0,1,0,0,0,0,0,0,0,-72,0,1,0,0,0,0,0,0,

%U 0,0,-90,0,1

%N A triangular sequence of coefficients based on an expansion of a Enneper like Sheffer expansion function: g(t) = t; f(t) = t; p(x,t) = Exp[x*(t)]*(1 - f(t)2).

%C Row sums:

%C {1, 1, -1, -5, -11, -19, -29, -41, -55, -71, -89};

%C Here is the relationship that seems to hold:

%C Weierstrass{f,g)-> Sheffer{g,fbar}.

%F g(t) = t; f(t) = t; p(x,t) = Exp[x*(t)]*(1 - f(t)2)=Sum(P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=n!*Coefficients(P(x,n)).

%e {1},

%e {0, 1},

%e {-2, 0, 1},

%e {0, -6, 0, 1},

%e {0, 0, -12, 0, 1},

%e {0, 0, 0, -20, 0, 1},

%e {0, 0, 0, 0, -30, 0, 1},

%e {0, 0, 0, 0, 0, -42, 0, 1},

%e {0, 0, 0, 0, 0, 0, -56, 0, 1},

%e {0, 0, 0, 0, 0, 0, 0, -72, 0, 1},

%e {0, 0, 0, 0, 0, 0, 0, 0, -90, 0, 1}

%t Clear[p, f, g] g[t_] = t; f[t] = t; p[t_] = Exp[x*g[t]]*(1 - f[t]^2); g = Table[ FullSimplify[ExpandAll[(n!)*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30}]], n], x], {n, 0, 10}]; Flatten[a]

%K uned,tabl,sign

%O 1,4

%A _Roger L. Bagula_, Apr 24 2008