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A209578 Triangle of coefficients of polynomials v(n,x) jointly generated with A209577; see the Formula section.

%I A209578 #7 Mar 30 2012 18:58:15
%S A209578 1,3,1,5,3,1,9,8,4,1,15,19,13,5,1,25,41,36,19,6,1,41,84,90,60,26,7,1,
%T A209578 67,165,210,169,92,34,8,1,109,315,465,439,287,133,43,9,1,177,588,990,
%U A209578 1073,818,454,184,53,10,1,287,1079,2043,2502,2178,1405,681,246
%N A209578 Triangle of coefficients of polynomials v(n,x) jointly generated with A209577; see the Formula section.
%C A209578 Alternating row sums: 1,0,2,1,3,2,4,3,5,4,... (A028242).
%C A209578 For a discussion and guide to related arrays, see A208510.
%F A209578 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209578 v(n,x)=(x+1)*u(n-1,x)+v(n-1,x)+1,
%F A209578 where u(1,x)=1, v(1,x)=1.
%e A209578 First five rows:
%e A209578 1
%e A209578 1...1
%e A209578 3...2....1
%e A209578 5...6....3....1
%e A209578 9...13...10...4...1
%e A209578 First three polynomials v(n,x): 1, 1 + x , 3 + 2x + x^2.
%t A209578 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209578 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209578 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
%t A209578 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209578 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209578 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209578 TableForm[cu]
%t A209578 Flatten[%]   (* A209577 *)
%t A209578 Table[Expand[v[n, x]], {n, 1, z}]
%t A209578 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209578 TableForm[cv]
%t A209578 Flatten[%]   (* A209578 *)
%Y A209578 Cf. A209577, A208510.
%K A209578 nonn,tabl
%O A209578 1,2
%A A209578 _Clark Kimberling_, Mar 11 2012