A001753 Expansion of 1/((1+x)*(1-x)^6).

1, 5, 16, 40, 86, 166, 296, 496, 791, 1211, 1792, 2576, 3612, 4956, 6672, 8832, 11517, 14817, 18832, 23672, 29458, 36322, 44408, 53872, 64883, 77623, 92288, 109088, 128248, 150008, 174624, 202368, 233529

Offset

0,2

Comments

Number of symmetric nonnegative integer 5 X 5 matrices with sum of elements equal to 4*n under action of dihedral group D_4.

a(n) = A108561(n+6,n) for n>0. - Reinhard Zumkeller, Jun 10 2005

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1)

Formula

a(n) = Sum{k=0..n} (-1)^(n-k)*binomial(k+5, 5); a(n) = (4*n^5 + 70*n^4 + 460*n^3 + 1400*n^2 + 1936*n + 945)/960 + (-1)^n/64. - Paul Barry, Jul 01 2003

a(n) = a(n-2) + (n*(n + 1)*(n + 2)*(n - 1))/24, a(1) = 0, a(2) = 1; (15*(-1)^n - 15*(-1)^(2*n) + 96*n - 160*(-1)^(2*n)*n + 200*n^2 - 200*(-1)^(2*n)*n^2 + 140*n^3 - 80*(-1)^(2*n)*n^3 + 40*n^4 - 10*(-1)^(2*n)*n^4 + 4*n^5)/960. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004

a(n) + a(n+1) = A000389(n+6). - R. J. Mathar, Mar 14 2011

Example

There are 5 symmetric nonnegative integer 5 X 5 matrices with sum of elements equal to 4 under action of D_4:
[1 0 0 0 1] [0 0 1 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0]
[0 0 0 0 0] [0 0 0 0 0] [0 1 0 1 0] [0 0 1 0 0] [0 0 0 0 0]
[0 0 0 0 0] [1 0 0 0 1] [0 0 0 0 0] [0 1 0 1 0] [0 0 4 0 0]
[0 0 0 0 0] [0 0 0 0 0] [0 1 0 1 0] [0 0 1 0 0] [0 0 0 0 0]
[1 0 0 0 1] [0 0 1 0 0] [0 0 0 0 0] [0 0 0 0 0] [0 0 0 0 0].

Mathematica

CoefficientList[Series[1/((1+x)*(1-x)^6), {x, 0, 50}], x] (* G. C. Greubel, Nov 22 2017 *)
LinearRecurrence[{5, -9, 5, 5, -9, 5, -1}, {1, 5, 16, 40, 86, 166, 296}, 40] (* Harvey P. Dale, Jun 05 2021 *)

Prog

(Magma) [(4*n^5+70*n^4+460*n^3+1400*n^2+1936*n+945)/960+(-1)^n/64: n in [0..40]]; // Vincenzo Librandi, Aug 15 2011
(PARI) a(n)=(4*n^5+70*n^4+460*n^3+1400*n^2+1936*n)\/960+1 \\ Charles R Greathouse IV, Apr 17 2012

Crossrefs

Cf. A000217, A002620, A008804, A038163, A054343, A001769 (partial sums), A001752 (first differences), A169793 (binomial transf).

Sequence in context: A027085 A099452 A006007 * A202087 A361074 A309403

Adjacent sequences: A001750 A001751 A001752 * A001754 A001755 A001756

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Comment and example from Vladeta Jovovic, May 14 2000

Status

approved