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A106607 Expansion of (1+t^3)^2/((1-t)*(1-t^2)^2*(1-t^4)). 4
1, 1, 3, 5, 9, 13, 20, 28, 39, 51, 67, 85, 107, 131, 160, 192, 229, 269, 315, 365, 421, 481, 548, 620, 699, 783, 875, 973, 1079, 1191, 1312, 1440, 1577, 1721, 1875, 2037, 2209, 2389, 2580, 2780, 2991, 3211, 3443, 3685, 3939, 4203, 4480, 4768, 5069, 5381, 5707, 6045 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Molien series for 5-dimensional group of order 8.
For of each of the quadrisections the n-th term is a polynomial in n of degree 3. - Ralf Stephan, Nov 16 2010
Number of non-isomorphic 3 X 3 nonnegative integer matrices with all row and column sums equal to n up to permutations of rows and columns. - Andrew Howroyd, Apr 08 2020
Take the square spiral on the square grid, with cells on the spiral numbered starting at 1. Every time the spiral crosses the x- or y-axis, calculate the sum of the numbers on the intersection of the spiral and the axis. This produces the present sequence (see illustration). - Karl-Heinz Hofmann, Aug 14 2022
LINKS
S. Ling and P. Solé, Type II Codes over F_4 + u F_4, European J. Combinatorics, 22 (2001), pp. 983-997.
FORMULA
G.f.: (1-x+x^2)^2/( (1+x)*(1+x^2)*(1-x)^4 ). - R. J. Mathar, Dec 18 2014
a(n) = (4*n^3 +18*n^2 +56*n +3*(9*(-1)^n +2*(1-i)*(-i)^n +2*(1+i)*i^n +19))/96 where i is the imaginary unit. - Colin Barker, Feb 08 2016
E.g.f.: (1/48)*(6*(cos(x) - sin(x)) + p(x)*sinh(x) + (27 + p(x))*cosh(x)), where p(x) = 15 + 39*x + 15*x^2 + 2*x^3. - G. C. Greubel, Sep 08 2021
EXAMPLE
The a(4) = 9 symmetric matrices are:
[0 0 4] [0 1 3] [0 1 3] [0 2 2] [0 2 2]
[0 4 0] [1 2 1] [1 3 0] [2 0 2] [2 1 1]
[4 0 0] [3 1 0] [3 0 1] [2 2 0] [2 1 1]
.
[1 1 2] [1 0 3] [1 1 2] [2 0 2]
[1 2 1] [0 4 0] [1 3 0] [0 4 0]
[2 1 1] [3 0 1] [2 0 2] [2 0 2]
MAPLE
(1+t^3)^2/((1-t)*(1-t^2)^2*(1-t^4));
seq(coeff(series(%, t, n+1), t, n), n=0..60);
MATHEMATICA
LinearRecurrence[{3, -3, 1, 1, -3, 3, -1}, {1, 1, 3, 5, 9, 13, 20}, 61] (* G. C. Greubel, Sep 08 2021 *)
PROG
(PARI) a(n) = i=I; (4*n^3+18*n^2+56*n+3*(9*(-1)^n+(2-2*i)*(-i)^n+(2+2*i)*i^n+19))/96 \\ Colin Barker, Feb 08 2016
(Sage)
def A106607_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x^3)^2/((1-x)*(1-x^2)^2*(1-x^4)) ).list()
A106607_list(60) # G. C. Greubel, Sep 08 2021
CROSSREFS
Row n=3 of A333737.
Cf. A100779.
Sequence in context: A108754 A033499 A267262 * A305082 A007042 A178415
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, May 12 2005
STATUS
approved

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Last modified March 29 06:34 EDT 2024. Contains 371265 sequences. (Running on oeis4.)