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A210793 Triangle of coefficients of polynomials u(n,x) jointly generated with A210794; see the Formula section. 3
1, 2, 1, 3, 4, 2, 6, 10, 8, 3, 9, 24, 27, 16, 5, 18, 51, 74, 62, 30, 8, 27, 108, 189, 200, 136, 56, 13, 54, 216, 450, 574, 488, 282, 102, 21, 81, 432, 1026, 1536, 1571, 1128, 569, 184, 34, 162, 837, 2268, 3864, 4598, 3967, 2486, 1118, 328, 55, 243, 1620 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Row n starts with A038754(n) and ends with F(n), where F=A000045 (Fibonacci numbers).
Row sums: A000244 (powers of 3).
Alternating row sums: A000012 (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, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 29 2012
LINKS
FORMULA
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = (x+2)*u(n-1,x) + (x-1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 29 2012: (Start)
As DELTA(triangle T(n,k) with 0 <= k <= n:
G.f.: (1 + x - y*x^2 - 2*y*x^2 - y^2*x^2)/(1 - y*x - 3*x^2 - 2*y*x^2 - y^2*x^2).
T(n,k) = T(n-1,k-1) + 3*T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k <= n.
EXAMPLE
First five rows:
1;
2, 1;
3, 4, 2;
6, 10, 8, 3;
9, 24, 27, 16, 5;
First three polynomials u(n,x):
1
2 + x
3 + 4x + 2x^2.
From Philippe Deléham, Mar 29 2012: (Start)
(1, 1, -1, -1, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
3, 4, 2, 0;
6, 10, 8, 3, 0;
9, 24, 27, 16, 5, 0;
18, 51, 74, 62, 30, 8, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c;
d[x_] := h + x; e[x_] := p + x;
v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f;
j = 1; c = 0; h = 2; p = -1; f = 0;
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[%] (* A210793 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210794 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000244 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000012 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A077925 *)
CROSSREFS
Sequence in context: A209137 A269752 A122164 * A281715 A076632 A105646
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 26 2012
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)