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A209149 Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.

%I A209149 #18 Jan 22 2020 03:35:49
%S A209149 1,3,1,6,5,1,12,16,7,1,24,44,30,9,1,48,112,104,48,11,1,96,272,320,200,
%T A209149 70,13,1,192,640,912,720,340,96,15,1,384,1472,2464,2352,1400,532,126,
%U A209149 17,1,768,3328,6400,7168,5152,2464,784,160,19,1,1536,7424
%N A209149 Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.
%C A209149 Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...
%C A209149 For a discussion and guide to related arrays, see A208510.
%C A209149 As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 08 2012
%C A209149 A skew triangle of A209144. - _Philippe Deléham_, Mar 08 2012
%C A209149 Riordan array ( (1 + x)/(1 - 2*x), x/(1 - 2*x) ). Cf. A118800. Matrix inverse is a signed version of A112626. - _Peter Bala_, Jul 17 2013
%F A209149 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209149 v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A209149 where u(1,x)=1, v(1,x)=1.
%F A209149 As DELTA-triangle:
%F A209149 T(n,k) = 2*T(n-1,k) + T(n-1,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 08 2012
%F A209149 As DELTA-triangle: G.f. is (1+x)/(1-2*x-yx). - _Philippe Deléham_, Mar 08 2012
%e A209149 First five rows:
%e A209149    1;
%e A209149    3,  1;
%e A209149    6,  5,  1;
%e A209149   12, 16,  7, 1;
%e A209149   24, 44, 30, 9, 1;
%e A209149 First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.
%e A209149 v(1,x) = 1
%e A209149 v(2,x) = 3 + x
%e A209149 v(3,x) = (3 + x)*(2 + x)
%e A209149 v(4,x) = (3 + x)*(2 + x)^2
%e A209149 v(5,x) = (3 + x)*(2 + x)^3
%e A209149 v(n,x) = (3 + x)*(2 + x)^(n-2)for n > 1. - _Philippe Deléham_, Mar 08 2012
%t A209149 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209149 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209149 v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A209149 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209149 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209149 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209149 TableForm[cu]
%t A209149 Flatten[%]    (* A209148 *)
%t A209149 Table[Expand[v[n, x]], {n, 1, z}]
%t A209149 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209149 TableForm[cv]
%t A209149 Flatten[%]    (* A209149 *)
%Y A209149 Cf. A209144, A209146, A209148, A208510.
%Y A209149 Cf. A084938, A112626.
%K A209149 nonn,tabl
%O A209149 1,2
%A A209149 _Clark Kimberling_, Mar 07 2012