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A008804 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)). 19
1, 2, 4, 6, 10, 14, 20, 26, 35, 44, 56, 68, 84, 100, 120, 140, 165, 190, 220, 250, 286, 322, 364, 406, 455, 504, 560, 616, 680, 744, 816, 888, 969, 1050, 1140, 1230, 1330, 1430, 1540, 1650, 1771, 1892, 2024, 2156, 2300, 2444, 2600, 2756, 2925, 3094, 3276, 3458 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

b(n)=a(n-3) is the number of asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to n, under action of dihedral group D_4(b(0)=b(1)=b(2)=0). G.f. for b(n) is x^3/((1-x)^2*(1-x^2)*(1-x^4)). - Vladeta Jovovic, May 07 2000

If the offset is changed to 5, this is the 2nd Witt transform of A004526 [Moree]. - R. J. Mathar, Nov 08 2008

a(n) is the number of partitions of 2*n into powers of 2 less than or equal to 2^3. First differs from A000123 at n=8. - Alois P. Heinz, Apr 02 2012

a(n) is the number of bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. For n=1 we have for example 2 such bracelets with 4 black beads and 4 white beads: BBBWBWWW and BBWBWBWW. - Herbert Kociemba, Nov 27 2016

a(n) is the also number of aperiodic bracelets with 4 black beads and n+3 white beads which have no reflection symmetry. This is equivalent to saying that a(n) is the (n+7)th element of the DHK[4] (bracelet, identity, unlabeled, 4 parts)  transform of 1, 1, 1, ... (see Bower's link about transforms). Thus, for n >= 1 , a(n) = (DHK[4] c)_{n+7}, where c = (1 : n >= 1). This is because every bracelet with 4 black beads and n+3 white beads which has no reflection symmetry must also be aperiodic. This statement is not true anymore if we have k black beads where k is even >= 6. - Petros Hadjicostas, Feb 24 2019

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

C. G. Bower, Transforms (2)

Petros Hadjicostas, The aperiodic version of Herbert Kociemba's formula for bracelets with no reflection symmetry, 2019.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 197

Pieter Moree, The formal series Witt transform, Discr. Math. no. 295 vol. 1-3 (2005) 143-160. [From R. J. Mathar, Nov 08 2008]

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).

FORMULA

For a formula for a(n) see A014557.

a(n) = (84 +85*n +24*n^2 +2*n^3 +12*A056594(n+3) +3*(-1)^n*(n+4))/96. - R. J. Mathar, Nov 08 2008

a(n) = 2*(Sum_{k=0..floor(n/2)} A002620(k+2)) - A002620(n/2+2)*(1+(-1)^n)/2. - Paul Barry, Mar 05 2009

G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)). - Jaume Oliver Lafont, Sep 20 2009

Euler transform of length 4 sequence [2, 1, 0, 1]. - Michael Somos, Feb 05 2011

a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Feb 05 2011

From Herbert Kociemba, Nov 27 2016: (Start)

More generally gf(k) is the g.f. for the number of bracelets without reflection symmetry with k black beads and n-k white beads.

gf(k): x^k/2 * ( (1/k)*Sum_{n|k} phi(n)/(1 - x^n)^(k/n) - (1 + x)/(1 -x^2)^floor(k/2 + 1) ). The g.f. here is gf(4)/x^7 because of the different offset. (End)

EXAMPLE

G.f. = 1 + 2*x + 4*x^2 + 6*x^3 + 10*x^4 + 14*x^5 + 20*x^6 + 26*x^7 + 35*x^8 + ...

There are 10 asymmetric nonnegative integer 2 X 2 matrices with sum of elements equal to 7 under action of D_4:

[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]

[1 6] [2 5] [3 4] [2 4] [3 3] [4 2] [5 1] [3 2] [4 1] [2 3]

MAPLE

seq(coeff(series(1/((1-x)^2*(1-x^2)*(1-x^4)), x, n+1), x, n), n = 0..60); # G. C. Greubel, Sep 12 2019

MATHEMATICA

LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 2, 4, 6, 10, 14, 20, 26}, 60] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

gf[x_, k_]:=x^k/2 (1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])-(1+x)/(1-x^2)^Floor[k/2+1]); CoefficientList[Series[gf[x, 4]/x^7, {x, 0, 60}], x] (* Herbert Kociemba, Nov 27 2016 *)

Table[(84 +12*(-1)^n +85*n +3*(-1)^n*n +24*n^2 +2*n^3 +12*Sin[n Pi/2])/96, {n, 0, 60}] (* Eric W. Weisstein, Oct 12 2017 *)

CoefficientList[Series[1/((1-x)^4*(1+x)^2*(1+x^2)), {x, 0, 60}], x] (* Eric W. Weisstein, Oct 12 2017 *)

PROG

(PARI) a(n)=(84+12*(-1)^n+6*I*((-I)^n-I^n)+(85+3*(-1)^n)*n+24*n^2 +2*n^3)/96 \\ Jaume Oliver Lafont, Sep 20 2009

(PARI) {a(n) = my(s = 1); if( n<-7, n = -8 - n; s = -1); if( n<0, 0, s * polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; /* Michael Somos, Feb 02 2011 */

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 60); Coefficients(R!( 1/((1-x)^2*(1-x^2)*(1-x^4)) )); // G. C. Greubel, Sep 12 2019

(Sage)

def A008804_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P(1/((1-x)^2*(1-x^2)*(1-x^4))).list()

A008804_list(60) # G. C. Greubel, Sep 12 2019

(GAP) a:=[1, 2, 4, 6, 10, 14, 20, 26];; for n in [9..60] do a[n]:=2*a[n-1] -2*a[n-3]+2*a[n-4]-2*a[n-5]+2*a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 12 2019

CROSSREFS

Cf. A002620, A005232, A014557, A032246, A032248, A053307.

Column k=3 of A181322. Column k = 4 of A180472 (but with different offset).

Sequence in context: A071425 A115065 A333574 * A001307 A322010 A322003

Adjacent sequences:  A008801 A008802 A008803 * A008805 A008806 A008807

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 12 03:14 EDT 2020. Contains 335658 sequences. (Running on oeis4.)