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%I #20 Aug 06 2015 10:23:23
%S 4,1,0,2,4,3,4,4,4,5,4,6,8,7,8,8,8,9,12,10,12,15,12,12,16,17,16,18,20,
%T 19,20,20,24,25,24,26,28,27,28,32,32,33,36,34,36,39,40,40,44,45,44
%N Based on an erroneous version of A008614.
%C This is based on the formula in Burnside, Section 267, at the foot of page 363. Unfortunately there is a typo in the formula - the term with numerator 21 should have denominator (1+x)(1-x^3). This produces a sequence with 4's in the denominators. Multiplying by 4 gives a sequence of integers, shown here. This is included in the OEIS in accordance with our policy of publishing incorrect sequences together with pointers to the correct versions. - _N. J. A. Sloane_, Feb 21 2012
%D W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955, section 267, page 363
%F A precise definition is: Take the generating function as given by Burnside, expand as a Taylor series, and multiply by 4.
%F Expansion of (-4 - x + 2 x^3 + x^4 - 2 x^5 - 2 x^6 + 2 x^7 + 3 x^8 + 2 x^9 - 3 x^11)/(-1 + x^3 (1 + x - x^7 - x^8 + x^11))
%t (* expansion*)
%t w = Exp[I*2*Pi/7];
%t p[x_] = FullSimplify[ExpandAll[(4/168)*(1/(1 - x)^3 + 21/((1 - x)*(1 - x^2)) + 42/((1 - x)*(1 + x^2)) + 56/(1 - x^3) + 24/((1 - w*x)*(1 - w^2*x)*(1 - w^4*x)) + 24/((1 - w^6*x)*(1 - x*w^5)*(1 - x*w^3)))]];
%t a = Table[SeriesCoefficient[Series[FullSimplify[ExpandAll[p[x]]], {x, 0, 50}], n], {n, 0, 50}]
%t (* recursion*)
%t b[1] = 4; b[2] = 1; b[3] = 0; b[4] = 2; b[5] = 4; b[6] = 3;
%t b[7] = 4; b[8] = 4; b[9] = 4; b[10] = 5; b[11] = 4;
%t b[n_Integer?Positive] :=
%t b[n] = -489 + 11 n + n^2 - b[-11 + n] - 3 b[-10 + n] - 6 b[-9 + n] -
%t 9 b[-8 + n] - 11 b[-7 + n] - 12 b[-6 + n] - 12 b[-5 + n] -
%t 11 b[-4 + n] - 9 b[-3 + n] - 6 b[-2 + n] - 3 b[-1 + n];
%t Table[b[n], {n, 1, Length[a]}]
%o (PARI) Vec((-4-x+2*x^3+x^4-2*x^5-2*x^6+2*x^7+3*x^8+2*x^9-3*x^11)/(-1+x^3*(1+x-x^7-x^8+x^11))+O(x^9)) \\ _Charles R Greathouse IV_, Feb 13 2012
%Y Cf. A008616.
%K dead
%O 0,1
%A _N. J. A. Sloane_, Feb 21 2012
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