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A119376
Second diagonal above the central terms of pendular trinomial triangle A119369, ignoring leading zeros.
8
1, 4, 16, 63, 248, 980, 3894, 15563, 62555, 252789, 1026623, 4188390, 17159382, 70570380, 291253664, 1205935204, 5008047097, 20854723702, 87064706122, 364334839028, 1527943938306, 6420911995109, 27033938458595
OFFSET
0,2
COMMENTS
Equals convolution of A119370 and A119375, which is the prior diagonal above the central terms of triangle A119369.
LINKS
FORMULA
G.f.: A(x) = B(x)^2*(G(x) - 1)/x^2 = B(x)^2*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).
G.f.: 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) where f(x) = sqrt(1-4*x-2*x^2+x^4). - G. C. Greubel, Mar 17 2021
MATHEMATICA
f[x_]:= Sqrt[1-4*x-2*x^2+x^4];
CoefficientList[Series[2*(1-2*x-x^2 -f[x])/(x^2*(1+2*x^3+x^4 +(1+x)^2*f[x])*(1+x^2 +f[x])), {x, 0, 30}], x] (* G. C. Greubel, Mar 17 2021 *)
PROG
(PARI) {a(n)=polcoeff(4/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))^2* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2, n)}
(Sage)
def f(x): return sqrt(1-4*x-2*x^2+x^4)
def A119376_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) ).list()
A119376_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-2*x^2+x^4) >;
Coefficients(R!( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) )); // G. C. Greubel, Mar 17 2021
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 17 2006
STATUS
approved