Search: id:A002619
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%I A002619 M0887 N0336
%S A002619 1,1,2,3,8,24,108,640,4492,36336,329900,3326788,36846288,444790512,
%T A002619 5811886656,81729688428,1230752346368,19760413251956,336967037143596,
%U A002619 6082255029733168,115852476579940152,2322315553428424200,48869596859895986108
%N A002619 Number of 2-colored patterns on an n X n board.
%C A002619 Also number of orbits in the set of circular permutations (up to rotation) 
               under cyclic permutation of the elements. - Michael Steyer (m.steyer(AT)osram.de), 
               Oct 06 2001
%D A002619 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002619 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002619 J. E. A. Steggall, On the numbers of patterns which can be derived from 
               certain elements, Mess. Math., 37 (1907), 56-61.
%D A002619 A. Vella, Pattern avoidance in permutations: linear and cyclic orders, 
               The Electronic J. of Combinatorics, 9(2), 2002-3, #R18.
%H A002619 T. D. Noe, <a href="b002619.txt">Table of n, a(n) for n=1..100</a>
%F A002619 Sum_{k|n} u(n, k)/(nk), where u(n, k) = A047918(n, k).
%F A002619 a(n)=(1/n^2)Sum[phi(p)^2*(n/p)!*p^(n/p)], where phi is Euler's totient 
               function (A000010) and summation is over all divisors of n. (see 
               the Vella reference, p. 31). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Aug 23 2005
%e A002619 n=6: {(123456)}, {(135462), (246513), (351624)} and {(124635), (235146), 
               (346251), (451362), (562413), (613524)} are 3 of the 24 orbits, consisting 
               of 1, 3 and 6 permutations, respectively.
%p A002619 with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(phi(div[j])^2*(n/
               div[j])!*div[j]^(n/div[j]),j=1..tau(n))/n^2 end: seq(a(n),n=1..23); 
               # (Deutsch) (Deutsch)
%Y A002619 Cf. A002618, A047916, A064852, A064649.
%Y A002619 Cf. A000010.
%Y A002619 Sequence in context: A038561 A055981 A120260 this_sequence A129202 A127905 
               A009224
%Y A002619 Adjacent sequences: A002616 A002617 A002618 this_sequence A002620 A002621 
               A002622
%K A002619 nonn,nice,easy
%O A002619 1,3
%A A002619 N. J. A. Sloane (njas(AT)research.att.com), C. L. Mallows (colinm(AT)research.avayalabs.com)

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