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A210221
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210596; see the Formula section.
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4
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1, 2, 3, 2, 5, 4, 4, 8, 10, 8, 8, 13, 20, 24, 16, 16, 21, 40, 52, 56, 32, 32, 34, 76, 116, 128, 128, 64, 64, 55, 142, 240, 312, 304, 288, 128, 128, 89, 260, 488, 688, 800, 704, 640, 256, 256, 144, 470, 964, 1496, 1856, 1984, 1600, 1408, 512, 512, 233, 840
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OFFSET
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1,2
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COMMENTS
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Row sums: even-indexed Fibonacci numbers.
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012
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LINKS
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FORMULA
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u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = u(n-1,x) + 2*x*v(n-1,x) [Corrected by Indranil Ghosh, May 27 2017]
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-1), T(0,0) = T(1,0) = 1, T(2,0) = 2, T(1,1) = T(2,1) = T(2,2) = 0, T(n,k) = 0 if k < 0 or if k >= n. (End)
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EXAMPLE
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First five rows:
1;
2;
3, 2;
5, 4, 4;
8, 10, 8, 8;
First three polynomials u(n,x):
1
2
3 + 2x.
(1, 1, -1, 0, 0, 0, ...) DELTA (0, 0, 2, 0, 0, ...) begins:
1;
1, 0;
2, 0, 0;
3, 2, 0, 0;
5, 4, 4, 0, 0;
8, 10, 8, 8, 0, 0;
13, 20, 24, 16, 16, 0, 0; (End)
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MATHEMATICA
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u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x];
v[n_, 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]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
With[{m = 10}, Rest[CoefficientList[CoefficientList[Series[(1-2*y*x)/(1-x-2*y*x-x^2+2*y*x^2), {x, 0, m}, {y, 0, m}], x], y]]]//Flatten (* G. C. Greubel, Dec 16 2018 *)
T[n_, k_]:= If[k < 0 || k > n, 0, T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k] - 2*T[n-2, k-1]]; T[1, 0] = 1 ; T[2, 0] = 2; T[2, 1] = 0; Join[{1}, Table[T[n, k], {n, 1, 10}, {k, 0, n-2}]//Flatten] (* G. C. Greubel, Dec 17 2018 *)
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PROG
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(Python)
from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + 2*x*v(n - 1, x)
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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