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A293933 Poincare series for invariant polynomial functions on the space of binary forms of degree 7. 13
1, 0, 1, 0, 4, 0, 10, 4, 18, 13, 35, 26, 62, 52, 97, 92, 153, 144, 229, 223, 325, 329, 456, 460, 624, 636, 826, 856, 1084, 1119, 1398, 1449, 1766, 1845, 2214, 2306, 2743, 2860, 3349, 3507, 4065, 4245, 4889, 5107, 5820, 6093, 6893, 7200, 8108 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Many of these Poincare series has every other term zero, in which case these zeros have been omitted.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

Andries Brouwer, Poincaré Series (See n=7)

FORMULA

a(n) = (11/8640)*n^4 + (11/1080)*n^3 + O(n^2). - Robert Israel, Oct 20 2017

EXAMPLE

The Poincare series is (1 - t^6 + 2t^8 - t^10 + 5t^12 + 2t^14 + 6t^16 + 2t^18 + 5t^20 - t^22 + 2t^24 - t^26 + t^32) / (1 - t^4)(1 - t^6)(1 - t^8)(1 - t^10)(1 - t^12)

MAPLE

(x^16-x^13+2*x^12-x^11+5*x^10+2*x^9+6*x^8+2*x^7+5*x^6-x^5+2*x^4-x^3+1)/(-x^2+1)/(-x^3+1)/(-x^4+1)/(-x^5+1)/(-x^6+1);

f := gfun:-rectoproc({-12*a(n) - 60*a(n+1) - 168*a(n+2) - 348*a(n+3) - 588*a(n+4) - 852*a(n+5) - 1080*a(n+6) - 1212*a(n+7) - 1212*a(n+8) - 1080*a(n+9) - 852*a(n+10) - 588*a(n+11) - 348*a(n+12) - 168*a(n+13) - 60*a(n+14) - 12*a(n+15) + 11*n^4 + 418*n^3 + 6433*n^2 + 46778*n + 136380, a(0) = 1, a(1) = 0, a(2) = 1, a(3) = 0, a(4) = 4, a(5) = 0, a(6) = 10, a(7) = 4, a(8) = 18, a(9) = 13, a(10) = 35, a(11) = 26, a(12) = 62, a(13) = 52, a(14) = 97, a(15) = 92, a(16) = 153}, a(n), remember):

map(f, [$0..100]); # Robert Israel, Oct 20 2017

MATHEMATICA

a = DifferenceRoot[Function[{a, n},

{-60*a[n + 1] - 168*a[n + 2] -

348*a[n + 3] - 588*a[n + 4] -

852*a[n + 5] - 1080*a[n + 6] -

1212*a[n + 7] - 1212*a[n + 8] -

1080*a[n + 9] - 852*a[n + 10] -

588*a[n + 11] - 348*a[n + 12] -

168*a[n + 13] - 60*a[n + 14] -

12*a[n + 15] - 12*a[n] + 11*n^4 +

418*n^3 + 6433*n^2 + 46778*n + 136380 == 0,

a[0] == 1, a[1] == 0, a[2] == 1,

a[3] == 0, a[4] == 4, a[5] == 0,

a[6] == 10, a[7] == 4, a[8] == 18,

a[9] == 13, a[10] == 35, a[11] == 26,

a[12] == 62, a[13] == 52, a[14] == 97}]];

Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 04 2019, after Robert Israel *)

CROSSREFS

For these Poincare series for d = 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 20, 24 see A097852, A293933, A097851, A293934, A293935, A293936, A293937, A293938, A293939, A293940, A293941, A293942, A293943 respectively.

Sequence in context: A184363 A331451 A164735 * A158976 A211243 A181626

Adjacent sequences:  A293930 A293931 A293932 * A293934 A293935 A293936

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Oct 20 2017

STATUS

approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)