|
|
A209136
|
|
Triangle of coefficients of polynomials v(n,x) jointly generated with A209135; see the Formula section.
|
|
3
|
|
|
1, 1, 3, 1, 8, 5, 1, 15, 23, 11, 1, 24, 66, 66, 21, 1, 35, 150, 240, 165, 43, 1, 48, 295, 678, 747, 404, 85, 1, 63, 525, 1631, 2547, 2157, 947, 171, 1, 80, 868, 3500, 7246, 8560, 5864, 2182, 341, 1, 99, 1356, 6888, 18126, 28018, 26592, 15318, 4929, 683
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 11 2012
|
|
LINKS
|
|
|
FORMULA
|
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)*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-y*x+y*x^2-2*y^2*x^2)/(1-2*x-y*x+x^2-y*x^2-2*y^2*x^2).
T(n,k) = 2*T(n-1,k) + 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)
|
|
EXAMPLE
|
First five rows:
1;
1, 3;
1, 8, 5;
1, 15, 23, 11;
1, 24, 66, 66, 21;
First three polynomials v(n,x):
1
1 + 3x
1 + 8x + 5x^2.
(1, 0, 2/3, 1/3, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, ...) begins:
1;
1, 0;
1, 3, 0;
1, 8, 5, 0;
1, 15, 23, 11, 0;
1, 24, 66, 66, 21, 0;
1, 35, 150, 240, 165, 43, 0; (End)
|
|
MATHEMATICA
|
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|