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A008804 Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)). 11
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

LINKS

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

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) = 7/8+n^3/48+n^2/4+85*n/96+A056594(n+3)/8+(-1)^n*(n+4)/32. - 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, n divides 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]

MATHEMATICA

LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 2, 4, 6, 10, 14, 20, 26}, 80] (* 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, 50}], 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, 20}] (* Eric W. Weisstein, Oct 12 2017 *)

CoefficientList[Series[1/((-1 + x)^4 (1 + x)^2 (1 + x^2)), {x, 0, 20}], 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 */

CROSSREFS

Cf. A014557, A005232, A053307, A002620.

Column k=3 of A181322.

Sequence in context: A094589 A071425 A115065 * A001307 A088932 A088954

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 October 20 23:39 EDT 2018. Contains 316405 sequences. (Running on oeis4.)