|
|
A209417
|
|
Triangle of coefficients of polynomials u(n,x) jointly generated with A209418; see the Formula section.
|
|
3
|
|
|
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, 1, 1, 13, 81, 229, 463, 423, 247, 1, 1, 16, 100, 412, 886, 1384, 1072, 502, 1, 1, 17, 140, 584, 1766, 3086, 3896, 2618, 1013, 1, 1, 20, 165, 900, 2850, 6744, 9942, 10494, 6213, 2036, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,5
|
|
COMMENTS
|
For a discussion and guide to related arrays, see A208510.
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
|
|
LINKS
|
|
|
FORMULA
|
u(n,x) = x*u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle T(n,k) with 0 <= k <= n:
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).
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)
|
|
EXAMPLE
|
First five rows:
1;
1, 1;
1, 4, 1;
1, 5, 11, 1;
1, 8, 18, 26, 1;
First three polynomials v(n,x):
1
1 + x
1 + 4x + x^2.
(1, 0, 2, -3, 0, 0, 0, ...) DELTA (0, 1, 0, 2, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 4, 1, 0;
1, 5, 11, 1, 0;
1, 8, 18, 26, 1, 0;
1, 9, 38, 56, 57, 1, 0; (End)
|
|
MATHEMATICA
|
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, 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]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
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 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|