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%I #34 Jun 29 2014 03:49:02
%S 1,1,2,4,6,8,13,17,23,31,40,50,65,79,97,119,143,169,203,237,277,323,
%T 373,427,492,558,633,717,807,903,1014,1128,1254,1392,1539,1695,1870,
%U 2050,2246,2458,2682,2918,3178,3446,3734,4042,4366,4706,5075,5455,5860
%N Number of nonisomorphic 2-generator p-groups of class at most 2 and order p^n.
%H Vincenzo Librandi, <a href="/A213506/b213506.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 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 a(n) = sum_{r+s+t=n, r >= s >= t >= 0}( (t+1)+(1/2)*min{t,r-s}*(2t+1-min{t,r-s} ).
%F G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)). - _Tani Akinari_, Jun 28 2014
%F 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 := proc(n)
%p a := 0 ;
%p for t from 0 to n do
%p for s from t to n-t do
%p r := n-s-t ;
%p if r >= s then
%p m := min(t,r-s) ;
%p a := a+t+1+m*(2*t+1-m)/2 ;
%p end if;
%p end do:
%p end do:
%p return a;
%p end proc:
%p seq(A213506(n),n=0..70) ; # _R. J. Mathar_, Jun 26 2012
%t 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 (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 (PARI) Vec( 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)) + O(x^100)) \\ _Michel Marcus_, Jun 28 2014
%K nonn,easy
%O 0,3
%A _Arturo Magidin_, Jun 12 2012
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