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A209417 Triangle of coefficients of polynomials u(n,x) jointly generated with A209418; see the Formula section.

%I A209417 #20 Jan 24 2020 03:29:19
%S A209417 1,1,1,1,4,1,1,5,11,1,1,8,18,26,1,1,9,38,56,57,1,1,12,51,142,159,120,
%T A209417 1,1,13,81,229,463,423,247,1,1,16,100,412,886,1384,1072,502,1,1,17,
%U A209417 140,584,1766,3086,3896,2618,1013,1,1,20,165,900,2850,6744,9942,10494,6213,2036,1
%N A209417 Triangle of coefficients of polynomials u(n,x) jointly generated with A209418; see the Formula section.
%C A209417 For a discussion and guide to related arrays, see A208510.
%C A209417 Subtriangle of the triangle given by (1, 0, 2, -3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 01 2012
%H A209417 G. C. Greubel, <a href="/A209417/b209417.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209417 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209417 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A209417 where u(1,x)=1, v(1,x)=1.
%F A209417 From _Philippe Deléham_, Apr 01 2012: (Start)
%F A209417 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209417 G.f.: (1+x-3*y*x-3*y*x^2+2*y^2*x^2)/(1-3*y*x-x^2-y*x^2+2*y^2*x^2).
%F A209417 T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) -2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209417 First five rows:
%e A209417   1;
%e A209417   1,  1;
%e A209417   1,  4,  1;
%e A209417   1,  5, 11,  1;
%e A209417   1,  8, 18, 26,  1;
%e A209417 First three polynomials v(n,x):
%e A209417   1
%e A209417   1 + x
%e A209417   1 + 4x + x^2.
%e A209417 From _Philippe Deléham_, Apr 01 2012: (Start)
%e A209417 (1, 0, 2, -3, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, ...) begins:
%e A209417   1;
%e A209417   1,  0;
%e A209417   1,  1,  0;
%e A209417   1,  4,  1,  0;
%e A209417   1,  5, 11,  1,  0;
%e A209417   1,  8, 18, 26,  1,  0;
%e A209417   1,  9, 38, 56, 57,  1,  0; (End)
%t A209417 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209417 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209417 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209417 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209417 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209417 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209417 TableForm[cu]
%t A209417 Flatten[%]    (* A209417 *)
%t A209417 Table[Expand[v[n, x]], {n, 1, z}]
%t A209417 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209417 TableForm[cv]
%t A209417 Flatten[%]    (* A209418 *)
%t A209417 CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - x^2 - y*x^2 + 2*y^2*x^2), {x,0,10}, {y,0,10}], x], y] // Flatten (* _G. C. Greubel_, Jan 03 2018 *)
%Y A209417 Cf. A209418, A208510.
%K A209417 nonn,tabl
%O A209417 1,5
%A A209417 _Clark Kimberling_, Mar 09 2012