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A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.

%I A208524 #6 Mar 30 2012 18:58:13
%S A208524 1,1,1,1,3,3,1,6,10,5,1,10,22,23,11,1,15,40,65,60,21,1,21,65,145,195,
%T A208524 137,43,1,28,98,280,490,518,322,85,1,36,140,490,1050,1484,1372,723,
%U A208524 171,1,45,192,798,2016,3570,4368,3447,1624,341,1,55,255,1230,3570
%N A208524 Triangle of coefficients of polynomials u(n,x) jointly generated with A208525; see the Formula section.
%C A208524 Alternating row sums: 1,0,1,0,1,0,1,0,...
%F A208524 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208524 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A208524 where u(1,x)=1, v(1,x)=1.
%e A208524 First five rows:
%e A208524 1
%e A208524 1...1
%e A208524 1...3....3
%e A208524 1...6....10...5
%e A208524 1...10...22...23...11
%e A208524 First five polynomials u(n,x):
%e A208524 1
%e A208524 1 + x
%e A208524 1 + 3x + 3x^2
%e A208524 1 + 6x + 10x^2 + 5x^3
%e A208524 1 + 10x + 22x^2 + 23x^3 + 11x^4
%t A208524 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208524 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208524 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A208524 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208524 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208524 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208524 TableForm[cu]
%t A208524 Flatten[%]  (* A208524 *)
%t A208524 Table[Expand[v[n, x]], {n, 1, z}]
%t A208524 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208524 TableForm[cv]
%t A208524 Flatten[%]  (* A208525 *)
%t A208524 Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
%t A208524 Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
%t A208524 Table[u[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
%t A208524 Table[v[n, x] /. x -> -1, {n, 1, z}](*alt. row sums*)
%Y A208524 Cf. A208525.
%K A208524 nonn,tabl
%O A208524 1,5
%A A208524 _Clark Kimberling_, Feb 29 2012