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A122447
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Central terms of pendular trinomial triangle A122445.
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8
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1, 0, 1, 2, 8, 28, 107, 418, 1676, 6848, 28418, 119444, 507440, 2175500, 9400207, 40895602, 178984212, 787503168, 3481278734, 15454765948, 68871993872, 307981243608, 1381569997998, 6215433403188, 28036071086296
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OFFSET
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0,4
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COMMENTS
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G.f.: A(x) = 1/(1+x - x*B(x)) = (1 + x*H(x))/(1+x) = 1 + x^2*F(x)/B(x), where B(x) is g.f. of A122446, H(x) is g.f. of A122448, F(x) is g.f. of A122450.
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LINKS
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FORMULA
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G.f. satisfies: A(x) = 1+2*x - 2*x*(3+x)*A(x) + x*(4+3*x)*A(x)^2.
G.f.: A(x) = ( 1 +6*x +2*x^2 - sqrt(1 -4*x -4*x^2 +4*x^4) )/( 2*x*(4+3*x) ).
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MATHEMATICA
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f[x_]:= Sqrt[1-4*x-4*x^2+4*x^4];
CoefficientList[Series[(1+6*x+2*x^2-f[x])/(2*x*(4+3*x)), {x, 0, 30}], x] (* G. C. Greubel, Mar 17 2021 *)
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PROG
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(PARI) {a(n)=polcoeff(2*(1+2*x)/(1+6*x+2*x^2+sqrt(1-4*x-4*x^2+4*x^4+x*O(x^n))), n)}
(Sage)
def f(x): return sqrt(1-4*x-4*x^2+4*x^4)
P.<x> = PowerSeriesRing(QQ, prec)
return P( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) ).list()
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-4*x^2+4*x^4) >;
Coefficients(R!( ( 1+6*x+2*x^2 -f(x) )/( 2*x*(4+3*x) ) )); // G. C. Greubel, Mar 17 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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