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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Many of these Poincaré 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 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

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 July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)