OFFSET
0,5
COMMENTS
Many of these Poincaré series have 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).
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,0,-2,-2,-1,0,1,2,2,0,0,-1,-1,-1,0,1).
FORMULA
a(n) = (11/8640)*n^4 + (11/1080)*n^3 + O(n^2). - Robert Israel, Oct 20 2017
a(n) = floor((11*n^4+88*n^3+768*n^2+3168*n+8640)/8640 - (n mod 2)*(n+1)*(n+3)*11/96 - (n mod 3)*n*5/54). - Hoang Xuan Thanh, Apr 20 2026
EXAMPLE
The Poincaré series is (1 - t^6 + 2*t^8 - t^10 + 5*t^12 + 2*t^14 + 6*t^16 + 2*t^18 + 5*t^20 - t^22 + 2*t^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);
# Alternative:
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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Oct 20 2017
STATUS
approved
