cp's OEIS Frontend

A210196 Triangle of coefficients of polynomials v(n,x) jointly generated with A210195; see the Formula section.

%I A210196 #21 Feb 22 2022 12:43:02
%S A210196 1,1,4,1,8,8,1,12,24,16,1,16,48,64,32,1,20,80,160,160,64,1,24,120,320,
%T A210196 480,384,128,1,28,168,560,1120,1344,896,256,1,32,224,896,2240,3584,
%U A210196 3584,2048,512,1,36,288,1344,4032,8064,10752,9216,4608,1024,1,40
%N A210196 Triangle of coefficients of polynomials v(n,x) jointly generated with A210195; see the Formula section.
%C A210196 Periodic alternating row sums: 1, -3, 1, -3, 1, -3, ...
%C A210196 For a discussion and guide to related arrays, see A208510.
%F A210196 u(n,x) = u(n-1,x) + v(n-1,x) + 1, v(n,x) = 2*x*u(n-1,x) + 2*x*v(n-1,x) + 1, where u(1,x)=1, v(1,x)=1.
%F A210196 Conjecture: T(n,0) = 1 and T(n,k) = 2^(k+1)*binomial(n-1,k) if k>0. - _Knud Werner_, Jan 10 2022
%e A210196 First five rows:
%e A210196   1;
%e A210196   1,  4;
%e A210196   1,  8,  8;
%e A210196   1, 12, 24, 16;
%e A210196   1, 16, 48, 64, 32;
%e A210196 First three polynomials v(n,x): 1, 1 + 4x, 1 + 8x + 8x^2.
%t A210196 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210196 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210196 v[n_, x_] := 2 x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A210196 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210196 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210196 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210196 TableForm[cu]
%t A210196 Flatten[%]   (* A210195 *)
%t A210196 Table[Expand[v[n, x]], {n, 1, z}]
%t A210196 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210196 TableForm[cv]
%t A210196 Flatten[%]  (* A210196 *)
%Y A210196 Cf. A134347, A210195, A208510.
%K A210196 nonn,tabl
%O A210196 1,3
%A A210196 _Clark Kimberling_, Mar 18 2012