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Triangle of coefficients of polynomials v(n,x) jointly generated with A209146; see the Formula section.
5

%I #18 Jan 22 2020 03:35:49

%S 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 70,13,1,192,640,912,720,340,96,15,1,384,1472,2464,2352,1400,532,126,

%U 17,1,768,3328,6400,7168,5152,2464,784,160,19,1,1536,7424

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

%C Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...

%C For a discussion and guide to related arrays, see A208510.

%C 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 A skew triangle of A209144. - _Philippe Deléham_, Mar 08 2012

%C 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 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),

%F v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,

%F where u(1,x)=1, v(1,x)=1.

%F As DELTA-triangle:

%F 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 As DELTA-triangle: G.f. is (1+x)/(1-2*x-yx). - _Philippe Deléham_, Mar 08 2012

%e First five rows:

%e 1;

%e 3, 1;

%e 6, 5, 1;

%e 12, 16, 7, 1;

%e 24, 44, 30, 9, 1;

%e First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.

%e v(1,x) = 1

%e v(2,x) = 3 + x

%e v(3,x) = (3 + x)*(2 + x)

%e v(4,x) = (3 + x)*(2 + x)^2

%e v(5,x) = (3 + x)*(2 + x)^3

%e v(n,x) = (3 + x)*(2 + x)^(n-2)for n > 1. - _Philippe Deléham_, Mar 08 2012

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

%t v[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A209148 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A209149 *)

%Y Cf. A209144, A209146, A209148, A208510.

%Y Cf. A084938, A112626.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 07 2012