A246575 - OEIS

%I #37 Aug 21 2018 08:22:45

%S 0,1,1,0,-1,-2,-2,-2,-1,0,1,2,3,3,3,3,2,1,0,-1,-2,-3,-4,-4,-4,-4,-4,

%T -3,-2,-1,0,1,2,3,4,5,5,5,5,5,5,4,3,2,1,0,-1,-2,-3,-4,-5,-6,-6,-6,-6,

%U -6,-6,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,7,7,7,7,7,7,7,6,5,4,3

%N Expansion of (Product_{r>=1} (1-x^r))*x^(k^2) / Product_{i=1..k} (1-x^i)^2 with k=1.

%H J. Fulman, <a href="http://dx.doi.org/10.1090/S0273-0979-01-00920-X">Random matrix theory over finite fields</a>, Bull. Amer. Math. Soc., 39 (No. 1, 2002), 51-85, MR1864086 (2002i:60012). See top of page 70, Eq. 1, with k=1.

%F G.f.: x*exp( Sum_{n>=1} x^n/n * (1 - 2*x^n)/(1 - x^n) ). - _Paul D. Hanna_, Dec 14 2015

%p fGL:=proc(k) local a,i,r;

%p a:=x^(k^2)/mul((1-x^i)^2,i=1..k);

%p a:=a*mul(1-x^r,r=1..101);

%p series(a,x,101);

%p seriestolist(%);

%p end;fGL(1);

%t nmax = 100; CoefficientList[Series[x*Exp[Sum[x^k/k * (1 - 2*x^k)/(1 - x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 17 2015 *)

%o (PARI) {a(n) = my(A=1); A = x*exp( sum(k=1, n+1, x^k/k * (1-2*x^k)/(1 - x^k) +x*O(x^n) ) ); polcoeff(A, n)}

%o for(n=0, 100, print1(a(n), ", ")) \\ _Paul D. Hanna_, Dec 14 2015

%Y k=0 gives A010815.

%Y Cf. A246575, A246576, A246577, A246578, A078616.

%K sign

%O 0,6

%A _N. J. A. Sloane_, Aug 31 2014