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A209830
Triangle of coefficients of polynomials u(n,x) jointly generated with A209831; see the Formula section.
3
1, 1, 2, 1, 5, 5, 1, 7, 18, 13, 1, 10, 35, 59, 34, 1, 12, 61, 147, 185, 89, 1, 15, 90, 302, 558, 564, 233, 1, 17, 129, 527, 1324, 1986, 1685, 610, 1, 20, 170, 854, 2653, 5350, 6761, 4957, 1597, 1, 22, 222, 1278, 4811, 12066, 20383, 22277, 14406, 4181, 1
OFFSET
1,3
COMMENTS
Each row begins with 1 and ends with an odd-indexed Fibonacci number.
Alternating row sums: 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, 1/2, -3/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 16 2012
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 with 0 <= k <= n: G.f.: (1+x-3*y*x-3*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
As DELTA-triangle: 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) = 2 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 16 2012
EXAMPLE
First five rows:
1;
1, 2;
1, 5, 5;
1, 7, 18, 13;
1, 10, 35, 59, 34;
First three polynomials u(n,x):
1
1 + 2x
1 + 5x + 5x^2.
From Philippe Deléham, Mar 16 2012: (Start)
(1, 0, 1/2, -3/2, 0, 0, ...) DELTA (0, 2, 1/2, 1/2, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 5, 5, 0;
1, 7, 18, 13, 0;
1, 10, 35, 59, 34, 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
Sequence in context: A111785 A304462 A021468 * A386886 A209695 A033282
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 13 2012
STATUS
approved