A164355 Expansion of (1 - x^2)^4 * (1 - x^5) / ((1 - x)^5 * (1 - x^4)^2) in powers of x.

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

Offset

0,2

Links

Matthew House, Table of n, a(n) for n = 0..10000

Michael Somos, Rational Function Multiplicative Coefficients, 2014.

Index entries for linear recurrences with constant coefficients, signature (2,-3,4,-3,2,-1).

Formula

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

Example

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 + ...

Mathematica

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 *)

Prog

(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

Crossrefs

Sequence in context: A296031 A344214 A314005 * A287322 A217585 A314006

Adjacent sequences: A164352 A164353 A164354 * A164356 A164357 A164358

Keyword

nonn,easy

Author

Michael Somos, Aug 13 2009

Status

approved