OFFSET
0,2
COMMENTS
If x is replaced by x^5, this is the Molien series for the Heisenberg group H(5).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).
FORMULA
G.f.: (1 + x + 21*x^2 + x^3 + x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Jun 08 2015
a(n) = (25/24)*n^4 + (25/12)*n^3 + (35/24)*n^2 + (5/12)*n + 1 = 1 + 5*n*(n+1)*(5*n^2 + 5*n + 2)/24 = 1 + 5*A006322(n). - R. J. Mathar, Nov 09 2018
E.g.f.: (1/24)*(24 + 120*x + 360*x^2 + 200*x^3 + 25*x^4)*exp(x). - G. C. Greubel, Mar 24 2022
MATHEMATICA
CoefficientList[Series[(1 +x +21x^2 +x^3 +x^4)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 08 2015 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 6, 41, 156, 426}, 40] (* Harvey P. Dale, Dec 01 2017 *)
PROG
(Magma) I:=[1, 6, 41, 156, 426]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Jun 08 2015
(Sage) [1 + 5*n*(n+1)*(5*n^2+5*n+2)/24 for n in (0..50)] # G. C. Greubel, Mar 24 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jun 07 2015
STATUS
approved