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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
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
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
Sequence in context: A340242 A116647 A063858 * A284367 A280328 A280384
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 13 2012
STATUS
approved

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Last modified June 23 13:32 EDT 2024. Contains 373648 sequences. (Running on oeis4.)