A054343 -id:A054343 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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].

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)

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

[(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

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 *)

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

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

Comment and example from Vladeta Jovovic, May 14 2000

A343875 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotations and reflections.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 3, 1, 0, 1, 4, 11, 3, 1, 0, 1, 8, 31, 24, 6, 1, 0, 1, 10, 84, 113, 55, 6, 1, 0, 1, 16, 198, 528, 410, 99, 10, 1, 0, 1, 20, 440, 2003, 2710, 1091, 181, 10, 1, 0, 1, 29, 904, 6968, 15233, 10488, 2722, 288, 15, 1, 0, 1, 35, 1766, 21593, 75258, 82704, 34399, 5806, 461, 15, 1

Offset: 0

Andrew Howroyd, May 06 2021

			Array begins:
=====================================================
n\k | 0  1   2    3     4      5       6        7
----+------------------------------------------------
  0 | 1  0   0    0     0      0       0        0 ...
  1 | 1  1   1    1     1      1       1        1 ...
  2 | 1  1   3    4     8     10      16       20 ...
  3 | 1  3  11   31    84    198     440      904 ...
  4 | 1  3  24  113   528   2003    6968    21593 ...
  5 | 1  6  55  410  2710  15233   75258   331063 ...
  6 | 1  6  99 1091 10488  82704  563864  3376134 ...
  7 | 1 10 181 2722 34399 360676 3235551 25387944 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Rows n=0..3 are A000007, A000012, A005232, A054343.
Columns 0..1 are A000012, A008805(n-1).
Cf. A054252 (binary case), A318795, A343097, A343874.

U(n,s) = {(s(1)^(n^2) + s(1)^(n%2)*(2*s(4)^(n^2\4) + s(2)^(n^2\2)) + 2*s(1)^n*s(2)^(n*(n-1)/2) + 2*(s(1)^(n%2)*s(2)^(n\2))^n )/8}
T(n,k)={polcoef(U(n,i->1/(1-x^i) + O(x*x^k)), k)}

A054771 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, up to rotational symmetry.

Original entry on oeis.org

1, 3, 13, 43, 129, 327, 761, 1619, 3238, 6098, 10974, 18930, 31550, 50930, 80030, 122666, 183999, 270525, 390755, 555205, 777287, 1073297, 1463583, 1972533, 2630044, 3471508, 4539660, 5884564, 7565868, 9652788, 12226860, 15381924

Offset: 0

Vladeta Jovovic, May 18 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row n=3 of A343874.

Cf. A008610, A054343.

G.f.: (x^8 - 2*x^7 + 6*x^6 + 2*x^5 + 2*x^4 + 2*x^3 + 6*x^2 - 2*x + 1)/((1 - x^4)^2*(1 - x^2)^2*(1 - x)^5).

cp's OEIS Frontend

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

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A343875 Array read by antidiagonals: T(n,k) is the number of n X n nonnegative integer matrices with sum of elements equal to k, up to rotations and reflections.

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PARI

A054771 Number of nonnegative integer 3 X 3 matrices with sum of elements equal to n, up to rotational symmetry.

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Author

Keywords

Links

Crossrefs

Formula