|
| |
|
|
A209831
|
|
Triangle of coefficients of polynomials v(n,x) jointly generated with A209830; see the Formula section.
|
|
3
|
|
|
|
1, 1, 3, 1, 5, 8, 1, 8, 20, 21, 1, 10, 41, 71, 55, 1, 13, 65, 176, 235, 144, 1, 15, 99, 338, 684, 744, 377, 1, 18, 135, 590, 1536, 2490, 2285, 987, 1, 20, 182, 926, 3031, 6382, 8651, 6865, 2584, 1, 23, 230, 1388, 5359, 14065, 24875, 29020, 20284, 6765
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,3
|
|
|
COMMENTS
|
Each row begins with 1 and ends with an even-indexed Fibonacci number.
Alternating row sums: signed powers of 2.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2012
|
|
|
LINKS
|
Table of n, a(n) for n=1..55.
|
|
|
FORMULA
|
u(n,x)=x*u(n-1,x)+(x+1)*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 :
T(n,k) = 3*T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) - 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) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 16 2012
As DELTA-triangle with 0<=k<=n : G.f.: (1+x-3*y*x-2*y*x^2+y^2*x^2)/(1-3*y*x-x^2-2*y*x^2+y^2*x^2). - Philippe Deléham, Mar 16 2012
|
|
|
EXAMPLE
|
Contribution from Philippe Deléham, Mar 16 2012: (Start)
First five rows:
1
1...3
1...5....8
1...8....20...21
1...10...41...71...55
First three polynomials v(n,x): 1, 1 + 3x , 1 + 5x + 8x^2.
(1, 0, -1/3, -2/3, 0, 0, ...) DELTA (0, 3, -1/3, 1/3, 0, 0, ...) begins:
1
1, 0
1, 3, 0
1, 5, 8, 0
1, 8, 20, 21, 0
1, 10, 41, 71, 55, 0. (End)
|
|
|
MATHEMATICA
|
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*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[%] (* A209830 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209831 *)
|
|
|
CROSSREFS
|
Cf. A209830, A208510.
Sequence in context: A208760 A116647 A063858 * A284367 A280328 A280384
Adjacent sequences: A209828 A209829 A209830 * A209832 A209833 A209834
|
|
|
KEYWORD
|
nonn,tabl
|
|
|
AUTHOR
|
Clark Kimberling, Mar 13 2012
|
|
|
STATUS
|
approved
|
| |
|
|
