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A136185 Number of metacyclic groups of order p^n, prime p >= 3. 3

%I #18 Sep 08 2022 08:45:32

%S 1,2,3,5,7,11,14,20,25,33,40,51,60,74,86,103,118,139,157,182,204,233,

%T 259,293,323,362,397,441,481,531,576,632,683,745,802,871,934,1010,

%U 1080,1163,1240,1331,1415,1514,1606,1713,1813,1929,2037,2162,2279,2413,2539

%N Number of metacyclic groups of order p^n, prime p >= 3.

%C For number of metacyclic groups of order 2^n see A136184.

%H Klaus Brockhaus, <a href="/A136185/b136185.txt">Table of n, a(n) for n=1..1000</a> [Values computed with MAGMA]

%H Jonathan Bloom, Nathan McNew, <a href="https://arxiv.org/abs/1908.03953">Counting pattern-avoiding integer partitions</a>, arXiv:1908.03953 [math.CO], 2019.

%H Steven Liedahl, <a href="http://dx.doi.org/10.1006/jabr.1996.0381">Enumeration of metacyclic p-groups</a>, J. Algebra 186 (1996), no. 2, 436-446.

%H MAGMA Computational Algebra System, V2.14-1, <a href="http://magma.maths.usyd.edu.au/magma/htmlhelp/text701.htm#6850">Metacyclic p-groups</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,-2,-1,2,1,-1).

%F G.f.: -x*(x^7 - 2*x^5 + x^3 + x^2 - x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + x + 1)).

%F a(n) = (n^3 + 12*n^2 + c*n + d)/72, where c = 12 or 3 for n even/odd, and d = 72, 56 or 64 for n = 0, 1 or 2 (mod 3), according to the Liedahl paper. - _M. F. Hasler_, Jan 13 2015

%e a(4) = 5 since there are five metacyclic groups of order p^4; they have invariants <4, 0, 0, 4, [ p^4 ], >, <1, 2, 1, 2, [], >, <1, 2, 2, 2, [], >, <2, 2, 2, 2, [], > and <1, 3, 1, 1, [], > resp. (for details see MAGMA link).

%t Rest@ CoefficientList[Series[-x (x^7 - 2 x^5 + x^3 + x^2 - x - 1)/((x - 1)^4*(x + 1)^2*(x^2 + x + 1)), {x, 0, 53}], x] (* _Michael De Vlieger_, Nov 04 2019 *)

%o (Magma) [ NumberOfMetacyclicPGroups(3, n): n in [1..53] ];

%o (PARI) A136185(n)=(((n+12)*n+[12,3][1+n%2])*n+[72,56,64][1+n%3])/72 \\ _M. F. Hasler_, Jan 13 2015

%Y Cf. A136184.

%K nonn

%O 1,2

%A _Klaus Brockhaus_, Dec 19 2007

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