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%I #26 Mar 25 2022 03:09:53
%S 1,6,41,156,426,951,1856,3291,5431,8476,12651,18206,25416,34581,46026,
%T 60101,77181,97666,121981,150576,183926,222531,266916,317631,375251,
%U 440376,513631,595666,687156,788801,901326,1025481,1162041,1311806,1475601,1654276,1848706,2059791,2288456,2535651
%N Expansion of (1 + x + 21*x^2 + x^3 + x^4)/(1 - x)^5.
%C If x is replaced by x^5, this is the Molien series for the Heisenberg group H(5).
%H G. C. Greubel, <a href="/A257602/b257602.txt">Table of n, a(n) for n = 0..1000</a>
%H Yang-Hui He and John McKay, <a href="http://arxiv.org/abs/1505.06742">Sporadic and Exceptional</a>, arXiv:1505.06742 [math.AG], 2015.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F G.f.: (1 + x + 21*x^2 + x^3 + x^4)/(1-x)^5.
%F 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
%F 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
%F 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
%t CoefficientList[Series[(1 +x +21x^2 +x^3 +x^4)/(1-x)^5, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 08 2015 *)
%t LinearRecurrence[{5,-10,10,-5,1},{1,6,41,156,426},40] (* _Harvey P. Dale_, Dec 01 2017 *)
%o (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
%o (Sage) [1 + 5*n*(n+1)*(5*n^2+5*n+2)/24 for n in (0..50)] # _G. C. Greubel_, Mar 24 2022
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jun 07 2015
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