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A164355
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Expansion of (1 - x^2)^4 * (1 - x^5) / ((1 - x)^5 * (1 - x^4)^2) in powers of x.
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1
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1, 5, 11, 15, 18, 25, 33, 35, 36, 45, 55, 55, 54, 65, 77, 75, 72, 85, 99, 95, 90, 105, 121, 115, 108, 125, 143, 135, 126, 145, 165, 155, 144, 165, 187, 175, 162, 185, 209, 195, 180, 205, 231, 215, 198, 225, 253, 235, 216, 245, 275, 255, 234, 265, 297, 275, 252
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OFFSET
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0,2
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LINKS
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FORMULA
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Euler transform of length 5 sequence [5, -4, 0, 2, -1].
a(n) = 5 * b(n) unless n=0 where b() is multiplicative with b(2) = 11/5, b(2^e) = 2^e * 9/10 if e>1, b(p^e) = p^e if p>2.
a(-n) = -a(n) for all n in Z unless n=0. a(4*n) = 18*n unless n=0. a(4*n + 2) = 11 * (2*n + 1). a(2*n + 1) = 5 * (2*n + 1).
G.f.: 1 + 5*x / (1 - x)^2 + x^2 / (1 + x^2)^2.
a(n) = 2*a(n-1) - 3*a(n-2) + 4*a(n-3) - 3*a(n-4) + 2*a(n-5) - a(n-6). - Matthew House, Feb 12 2017
a(n) = (20 - (-i)^n - i^n)*n/4 for n>0, where i=sqrt(-1). - Colin Barker, Feb 12 2017
Sum_{k=1..n} a(k) ~ 5 * n^2 / 2. - Amiram Eldar, Dec 30 2023
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EXAMPLE
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G.f. = 1 + 5*x + 11*x^2 + 15*x^3 + 18*x^4 + 25*x^5 + 33*x^6 + 35*x^7 + 36*x^8 + ...
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MATHEMATICA
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a[ n_] := Boole[n == 0] + 5 n - If[ EvenQ[n], (-1)^(n/2) n/2, 0]; (* Michael Somos, Apr 17 2015 *)
a[ n_] := (-1)^Boole[n < 0] SeriesCoefficient[ (1 - x^2)^4 (1 - x^5) / ((1 - x)^5 (1 - x^4)^2), {x, 0, Abs@n}]; (* Michael Somos, Jan 07 2019 *)
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PROG
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(PARI) {a(n) = (n == 0) + 5*n - if( n%2==0, (-1)^(n/2) * n/2, 0)};
(PARI) Vec((1 + x)^2 * (1 + x + x^2 + x^3 + x^4) / ((1 - x)^2*(1 + x^2)^2) + O(x^60)) \\ Colin Barker, Feb 12 2017
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1 + 5*x/(1-x)^2 + x^2/(1+x^2)^2)); // G. C. Greubel, Sep 25 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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