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

1, 1, 2, 4, 6, 8, 13, 17, 23, 31, 40, 50, 65, 79, 97, 119, 143, 169, 203, 237, 277, 323, 373, 427, 492, 558, 633, 717, 807, 903, 1014, 1128, 1254, 1392, 1539, 1695, 1870, 2050, 2246, 2458, 2682, 2918, 3178, 3446, 3734, 4042, 4366, 4706, 5075, 5455, 5860

Offset: 0

Author

Arturo Magidin, Jun 12 2012

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• A. Ahmad, A. Magidin and R. F. Morse, Two generator p-groups of nilpotency class 2 and their conjugacy classes, Publ. Math. Debrecen 81 (2012), no. 1-2, 145-166.
• C. Voll, Enumerating finite class-2-nilpotent groups on 2 generators, C. R. Math. Acad. Sci. Paris 347 (2009), no. 23-24, 1347-1350.

Programs

Maple

A213506 := proc(n)
        a := 0 ;
        for t from 0 to n do
                for s from t to n-t do
                        r := n-s-t ;
                        if r >= s then
                                m := min(t,r-s) ;
                                a := a+t+1+m*(2*t+1-m)/2 ;
                        end if;
                end do:
        end do:
        return a;
end proc:
seq(A213506(n),n=0..70) ; # R. J. Mathar, Jun 26 2012

Mathematica

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 *)

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

PARI

Vec( 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jun 28 2014

Formula

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}) ).

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)^2*(1-x^4)). - Tani Akinari, Jun 28 2014

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]