login
A208759
Triangle of coefficients of polynomials u(n,x) jointly generated with A208760; see the Formula section.
3
1, 1, 2, 1, 4, 6, 1, 6, 16, 16, 1, 8, 30, 56, 44, 1, 10, 48, 128, 188, 120, 1, 12, 70, 240, 504, 608, 328, 1, 14, 96, 400, 1080, 1872, 1920, 896, 1, 16, 126, 616, 2020, 4512, 6672, 5952, 2448, 1, 18, 160, 896, 3444, 9352, 17856, 23040, 18192, 6688, 1, 20, 198, 1248, 5488, 17472, 40600, 67776, 77616, 54976, 18272
OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012
FORMULA
u(n,x) = u(n-1,x) + 2*x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + 2*x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 18 2012: (Start)
As DELTA-triangle with 0 <= k <= n:
G.f.: (1-2y*x-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
1, 2;
1, 4, 6;
1, 6, 16, 16;
1, 8, 30, 56, 44;
First five polynomials u(n,x):
1
1 + 2x
1 + 4x + 6x^2
1 + 6x + 16x^2 + 16x^3
1 + 8x + 30x^2 + 56x^3 + 44x^4
From Philippe Deléham, Mar 18 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 2, 1, -1, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 4, 6, 0;
1, 6, 16, 16, 0;
1, 8, 30, 56, 44, 0;
1, 10, 48, 128, 188, 120, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A208759 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208760 *)
Rest[CoefficientList[CoefficientList[Series[(1-2*y*x-2*y^2*x^2)/(1-x-2*y*x- 2*y^2*x^2), {x, 0, 20}, {y, 0, 20}], x], y]//Flatten] (* G. C. Greubel, Mar 28 2018 *)
CROSSREFS
Sequence in context: A208915 A199704 A062344 * A033877 A059369 A369518
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 02 2012
EXTENSIONS
Terms a(58) onward added by G. C. Greubel, Mar 28 2018
STATUS
approved