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A209132 Triangle of coefficients of polynomials v(n,x) jointly generated with A209131; see the Formula section.

%I A209132 #15 Jan 24 2020 03:26:24
%S A209132 1,0,3,0,4,5,0,4,12,11,0,4,20,36,21,0,4,28,76,92,43,0,4,36,132,244,
%T A209132 228,85,0,4,44,204,508,716,540,171,0,4,52,292,916,1732,1972,1252,341,
%U A209132 0,4,60,396,1500,3564,5436,5196,2844,683,0,4,68,516,2292,6564
%N A209132 Triangle of coefficients of polynomials v(n,x) jointly generated with A209131; see the Formula section.
%C A209132 For a discussion and guide to related arrays, see A208510.
%C A209132 As triangle T(n,k) with 0 <= k <= n, it is (0, 4/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 21 2012
%C A209132 Row sums are powers of 3 (A000244). - _Philippe Deléham_, Mar 21 2012
%F A209132 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
%F A209132 v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
%F A209132 where u(1,x)=1, v(1,x)=1.
%F A209132 T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = 1, T(2,0) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k >= n.- _Philippe Deléham_, Mar 21 2012
%F A209132 G.f.: (-1-2*x*y+x)*x*y/((1+x*y)*(2*x*y+x-1)). - _R. J. Mathar_, Aug 12 2015
%e A209132 First five rows:
%e A209132   1;
%e A209132   0,  3;
%e A209132   0,  4,  5;
%e A209132   0,  4, 12, 11;
%e A209132   0,  4, 20, 36, 21;
%e A209132 First three polynomials v(n,x):
%e A209132   1
%e A209132      3x
%e A209132      4x + 5x^2.
%t A209132 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209132 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209132 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A209132 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209132 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209132 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209132 TableForm[cu]
%t A209132 Flatten[%]    (* A209131 *)
%t A209132 Table[Expand[v[n, x]], {n, 1, z}]
%t A209132 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209132 TableForm[cv]
%t A209132 Flatten[%]    (* A209132 *)
%Y A209132 Cf. A209131, A208510.
%K A209132 nonn,tabl
%O A209132 1,3
%A A209132 _Clark Kimberling_, Mar 05 2012