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A213506 Number of nonisomorphic 2-generator p-groups of class at most 2 and order p^n.

%I A213506 #38 Feb 19 2025 01:01:31
%S A213506 1,1,2,4,6,8,13,17,23,31,40,50,65,79,97,119,143,169,203,237,277,323,
%T A213506 373,427,492,558,633,717,807,903,1014,1128,1254,1392,1539,1695,1870,
%U A213506 2050,2246,2458,2682,2918,3178,3446,3734,4042,4366,4706,5075,5455,5860
%N A213506 Number of nonisomorphic 2-generator p-groups of class at most 2 and order p^n.
%H A213506 Vincenzo Librandi, <a href="/A213506/b213506.txt">Table of n, a(n) for n = 0..1000</a>
%H A213506 A. Ahmad, A. Magidin and R. F. Morse, <a href="http://www.ucs.louisiana.edu/~avm1260/preprints/ammpaper.pdf">Two generator p-groups of nilpotency class 2 and their conjugacy classes</a>, Publ. Math. Debrecen 81 (2012), no. 1-2, 145-166.
%H A213506 C. Voll, <a href="http://arxiv.org/abs/0908.1355">Enumerating finite class-2-nilpotent groups on 2 generators</a>, C. R. Math. Acad. Sci. Paris 347 (2009), no. 23-24, 1347-1350.
%F A213506 a(n) = Sum_{r+s+t=n, r >= s >= t >= 0}( (t+1)+(1/2)*min{t,r-s}*(2*t+1-min{t,r-s}) ).
%F A213506 G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)). - _Tani Akinari_, Jun 28 2014
%F A213506 a(n) = floor( (n^4+26*n^3+234*n^2+(909-64*(n mod 3))*n+1701)/1728+(n+1)*(-1)^n/64 ). [_Tani Akinari_, Jun 28 2014 - see PARI code]
%p A213506 A213506 := proc(n)
%p A213506         a := 0 ;
%p A213506         for t from 0 to n do
%p A213506                 for s from t to n-t do
%p A213506                         r := n-s-t ;
%p A213506                         if r >= s then
%p A213506                                 m := min(t,r-s) ;
%p A213506                                 a := a+t+1+m*(2*t+1-m)/2 ;
%p A213506                         end if;
%p A213506                 end do:
%p A213506         end do:
%p A213506         return a;
%p A213506 end proc:
%p A213506 seq(A213506(n),n=0..70) ; # _R. J. Mathar_, Jun 26 2012
%t A213506 CoefficientList[Series[1/((1 - x)*(1 - x^2)*(1 - x^3)^2*(1 - x^4)), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Jun 28 2014 *)
%o A213506 (PARI) a(n)=floor((n^4+26*n^3+234*n^2+(909-64*(n%3))*n+1701)/1728+(n+1)*(-1)^n/64) \\ _Tani Akinari_, Jun 28 2014
%o A213506 (PARI) Vec( 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)) + O(x^100)) \\ _Michel Marcus_, Jun 28 2014
%K A213506 nonn,easy
%O A213506 0,3
%A A213506 _Arturo Magidin_, Jun 12 2012