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A025170
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Expansion of g.f.: 1/(1 + 2*x + 9*x^2).
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3
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1, -2, -5, 28, -11, -230, 559, 952, -6935, 5302, 51811, -151340, -163619, 1689298, -1906025, -11391632, 39937489, 22649710, -404736821, 605626252, 2431378885, -10313394038, -1255621889, 95331790120, -179362983239, -499260144602, 2612787138355, -732232975292
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OFFSET
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0,2
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COMMENTS
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Reciprocal Chebyshev polynomial of second kind evaluated at 3 multiplied by (-1)^n.
From Sharon Sela (sharonsela(AT)hotmail.com), Jan 19 2002: (Start)
a(n) is (-1)^n times the determinant of the following tridiagonal n X n matrix:
[2 3 0 . . . . . . .]
[3 2 3 0 . . . . . .]
[0 3 2 3 0 . . . . .]
[. 0 3 2 3 0 . . . .]
[. . . . . . . . . .]
[. . . . . . . . . .]
[. . . . 0 3 2 3 0 .]
[. . . . . 0 3 2 3 0]
[. . . . . . 0 3 2 3]
[. . . . . . . 0 3 2]
(End)
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LINKS
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FORMULA
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a(n) = 3^n * ChebyshevU(n, -1/3).
a(n) = -(2*a(n-1) + 9*a(n-2)) for n>1, with a(0)=1, a(1)=-2. - Philippe Deléham, Sep 19 2009
a(n) = (-2)^n*Product_{k=1..n}(1 + 3*cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
E.g.f.: (1/4)*exp(-x)*(4*cos(2*sqrt(2)*x) - sqrt(2)*sin(2*sqrt(2)*x)). (End)
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MATHEMATICA
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Table[3^n ChebyshevU[n, -1/3], {n, 0, 24}]
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PROG
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(PARI) a(n)=if(n<0, 0, polcoeff(1/(1+2*x+9*x^2)+x*O(x^n), n))
(PARI) a(n)=if(n<0, 0, 3^n*subst(poltchebi(n+1)+3*poltchebi(n), 'x, -1/3)*3/8) /* Michael Somos, Sep 15 2005 */
(PARI) a(n)=if(n<0, 0, (-1)^n*matdet(matrix(n, n, i, j, if(abs(i-j)<2, 2+abs(i-j))))) /* Michael Somos, Sep 15 2005 */
(Magma) [(-3)^n*Evaluate(ChebyshevU(n+1), 1/3): n in [0..50]]; // G. C. Greubel, Jan 02 2024
(SageMath) [3^n*chebyshev_U(n, -1/3) for n in range(41)] # G. C. Greubel, Jan 02 2024
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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