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A208758 Triangle of coefficients of polynomials v(n,x) jointly generated with A208757; see the Formula section. 3
1, 0, 3, 0, 1, 8, 0, 1, 4, 22, 0, 1, 4, 16, 60, 0, 1, 4, 18, 56, 164, 0, 1, 4, 20, 68, 188, 448, 0, 1, 4, 22, 80, 248, 608, 1224, 0, 1, 4, 24, 92, 312, 864, 1920, 3344, 0, 1, 4, 26, 104, 380, 1152, 2928, 5952, 9136, 0, 1, 4, 28, 116, 452, 1472, 4128, 9696, 18192 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 02 2012
LINKS
FORMULA
u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = x*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k), 0 <= k <= n: g.f.: (1-x-y*x)/(1-(1+2*y)*x -2*y(y-1)*x^2). - Philippe Deléham, Mar 02 2012
As triangle T(n,k), 0 <= k <= n: T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - 2*T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 02 2012
EXAMPLE
First five rows:
1;
0, 3;
0, 1, 8;
0, 1, 4, 22;
0, 1, 4, 16, 60;
First five polynomials v(n,x):
1
3x
x + 8x^2
x + 4x^2 + 22x^3
x + 4x^2 + 16x^3 + 60^x4
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := x*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[%] (* A208757 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208758 *)
CROSSREFS
Sequence in context: A216806 A290776 A172249 * A320161 A291763 A161129
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 02 2012
STATUS
approved

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Last modified March 28 16:58 EDT 2024. Contains 371254 sequences. (Running on oeis4.)