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A115144
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Fifth convolution of A115140.
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5
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1, -5, 5, 0, 0, -1, -5, -20, -75, -275, -1001, -3640, -13260, -48450, -177650, -653752, -2414425, -8947575, -33266625, -124062000, -463991880, -1739969550, -6541168950, -24647883000, -93078189750, -352207870014, -1335293573130, -5071418015120, -19293438101000
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..1669
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FORMULA
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O.g.f.: 1/c(x)^5 = P(6, x) - x*P(5, x)*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(6, x)=1-4*x+3*x^2 and P(5, x)=1-3*x+x^2.
a(n) = -C5(n-5), n>=5, with C5(n) = A000344(n+2) (fifth convolution of Catalan numbers). a(0)=1, a(1)=-5, a(2)=5, a(3)=0=a(4). [1, -5, 5] is row n=5 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-5)*a(n) -2*(n-3)*(2*n-7)*a(n-1)=0. - R. J. Mathar, Sep 23 2021
From Peter Bala, Mar 05 2023: (Start)
a(n) = binomial(2*n - 6, n) - binomial(2*n - 6, n + 1).
a(n) = = -5/(n - 5)*binomial(2*n - 6, n) for n != 5.
a(n) = -A000344(n-3) for n >= 5. (End)
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MATHEMATICA
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CoefficientList[Series[(1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)
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PROG
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(PARI) my(x='x+O('x^30)); Vec((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-5*x+5*x^2 +(1-3*x+x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(Sage) ((1-5*x+5*x^2 +(1-3*x+x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019
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CROSSREFS
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Cf. A000344, A115139 - A115143, A115145 - A115149.
Sequence in context: A197738 A189232 A247667 * A200506 A285070 A285288
Adjacent sequences: A115141 A115142 A115143 * A115145 A115146 A115147
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KEYWORD
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sign,easy
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AUTHOR
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Wolfdieter Lang, Jan 13 2006
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STATUS
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approved
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