cp's OEIS Frontend

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

%I A209418 #21 Jan 24 2020 03:28:43
%S A209418 1,1,3,1,4,7,1,7,13,15,1,8,30,38,31,1,11,42,104,103,63,1,12,69,178,
%T A209418 321,264,127,1,15,87,331,657,921,649,255,1,16,124,484,1354,2200,2512,
%U A209418 1546,511,1,19,148,760,2266,4978,6856,6598,3595,1023,1,20,195,1020,3870,9384,16938,20226,16827,8204,2047
%N A209418 Triangle of coefficients of polynomials v(n,x) jointly generated with A209417; see the Formula section.
%C A209418 Alternating row sums:  signed powers of 2.
%C A209418 For a discussion and guide to related arrays, see A208510.
%C A209418 Subtriangle of the triangle given by (1, 0, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 01 2012
%H A209418 G. C. Greubel, <a href="/A209418/b209418.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A209418 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209418 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A209418 where u(1,x)=1, v(1,x)=1.
%F A209418 From _Philippe Deléham_, Apr 01 2012: (Start)
%F A209418 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A209418 G.f.: (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).
%F A209418 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) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A209418 First five rows:
%e A209418   1;
%e A209418   1,  3;
%e A209418   1,  4,  7;
%e A209418   1,  7, 13, 15;
%e A209418   1,  8, 30, 38, 31;
%e A209418 First three polynomials v(n,x):
%e A209418   1
%e A209418   1 + 3x
%e A209418   1 + 4x + 7x^2.
%e A209418 From _Philippe Deléham_, Apr 01 2012: (Start)
%e A209418 (1, 0, -2/3, -1/3, 0, 0, 0, ...) DELTA (0, 3, -2/3, 2/3, 0, 0, 0, ...) begins:
%e A209418   1;
%e A209418   1,  0;
%e A209418   1,  3,  0;
%e A209418   1,  4,  7,  0;
%e A209418   1,  7, 13, 15,  0;
%e A209418   1,  8, 30, 38, 31,  0; (End)
%t A209418 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209418 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209418 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A209418 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209418 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209418 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209418 TableForm[cu]
%t A209418 Flatten[%]    (* A209417 *)
%t A209418 Table[Expand[v[n, x]], {n, 1, z}]
%t A209418 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209418 TableForm[cv]
%t A209418 Flatten[%]    (* A209418 *)
%t A209418 CoefficientList[CoefficientList[Series[(1 + x)/(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 A209418 Cf. A209417, A208510.
%K A209418 nonn,tabl
%O A209418 1,3
%A A209418 _Clark Kimberling_, Mar 09 2012