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A210565
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Triangle of coefficients of polynomials u(n,x) jointly generated with A210595; see the Formula section.
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3
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1, 2, 2, 3, 5, 3, 4, 9, 10, 5, 5, 14, 22, 20, 8, 6, 20, 40, 51, 38, 13, 7, 27, 65, 105, 111, 71, 21, 8, 35, 98, 190, 256, 233, 130, 34, 9, 44, 140, 315, 511, 594, 474, 235, 55, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 11, 65, 255, 726, 1554, 2534, 3130, 2860, 1836, 744, 144
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OFFSET
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1,2
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COMMENTS
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Row n starts with n and ends with F(n+1), where F=A000045 (Fibonacci numbers).
Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0, ...
For a discussion and guide to related arrays, see A208510.
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LINKS
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FORMULA
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u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x) + 1,
v(n,x) = x*u(n-1,x) + v(n-1,x) + 1,
where u(1,x) = 1, v(1,x) = 1.
T(n, k) = [x^k]( u(n, x) ), where u(n, x) = (1+x)*u(n-1,x) + x^2*u(n-2,x) + 1 + x, u(1, x) = 1, and u(2, x) = 2 + 2*x. - G. C. Greubel, May 24 2021
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EXAMPLE
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First five rows:
1;
2, 2;
3, 5, 3;
4, 9, 10, 5;
5, 14, 22, 20, 8;
First three polynomials u(n,x):
u(1, x) = 1;
u(2, x) = 2 + 2*x;
u(3, x) = 3 + 5*x + 3*x^2.
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MATHEMATICA
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(* First program *)
u[1, x_]:= 1; v[1, x_]:= 1; z = 16;
u[n_, x_]:= x*u[n-1, x] + (x+1)*v[n-1, x] + 1;
v[n_, x_]:= x*u[n-1, x] + v[n-1, x] + 1;
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]
(* Second program *)
u[n_, x_]:= u[n, x]= If[n<2, (n+1)*(1+x)^n, (1+x)*u[n-1, x] +x^2*u[n-2, x] +1+x];
T[n_]:= CoefficientList[Series[u[n, x], {x, 0, n}], x];
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PROG
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(Sage)
@CachedFunction
def u(n, x): return (n+1)*(1+x)^n if (n<2) else (1+x)*u(n-1, x) + x^2*u(n-2, x) +1+x
def T(n): return taylor( u(n, x) , x, 0, n).coefficients(x, sparse=False)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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