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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

G. C. Greubel, Table of n, a(n) for n = 0..1000

S. Ling and P. Solé, Type II Codes over F_4 + u F_4, European J. Combinatorics, 22 (2001), pp. 983-997.

Karl-Heinz Hofmann, An alternative way to get the terms of A106607. Examples of a(1..18)

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

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

Adjacent sequences:  A106604 A106605 A106606 * A106608 A106609 A106610

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, May 12 2005

STATUS

approved

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Last modified September 27 19:37 EDT 2022. Contains 357063 sequences. (Running on oeis4.)