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A293933
Poincaré 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
OFFSET
0,5
COMMENTS
Many of these Poincaré series have every other term zero, in which case these zeros have been omitted.
LINKS
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 Poincaré 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 Poincaré 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 * A345057 A158976 A211243
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 20 2017
STATUS
approved