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A008804
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Expansion of 1/((1-x)^2*(1-x^2)*(1-x^4)).
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9
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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
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OFFSET
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0,2
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COMMENTS
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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]. [From 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
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LINKS
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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 to sequences with linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
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FORMULA
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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. [From R. J. Mathar, Nov 08 2008]
a(n)=2*sum{k=0..floor(n/2), A0002620(k+2)}-A0002620(n/2+2)(1+(-1)^n)/2. [From Paul Barry, Mar 05 2009]
G.f.: 1/((1-x)^4*(1+x)^2*(1+x^2)) [From Jaume Oliver Lafont, Sep 20 2009]
Euler transform of length 4 sequence [ 2, 1, 0, 1]. - Michael Somos Feb 05 2011
a(-8 - n) = -a(n). - Michael Somos Feb 05 2011
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EXAMPLE
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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]
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MAPLE
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1/((1-x)^2*(1-x^2)*(1-x^4));
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MATHEMATICA
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LinearRecurrence[{2, 0, -2, 2, -2, 0, 2, -1}, {1, 2, 4, 6, 10, 14, 20, 26}, 80] (* From Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
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PROG
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(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 [From Jaume Oliver Lafont, Sep 20 2009]
{a(n) = local(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 */
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CROSSREFS
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Cf. A014557, A005232, A053307.
Column k=3 of A181322.
Sequence in context: A094589 A071425 A115065 * A001307 A088932 A088954
Adjacent sequences: A008801 A008802 A008803 * A008805 A008806 A008807
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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