A246575 Expansion of (Product_{r>=1} (1-x^r))*x^(k^2) / Product_{i=1..k} (1-x^i)^2 with k=1.

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, -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, -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

Offset: 0

Author

N. J. A. Sloane, Aug 31 2014

Keywords

• sign

Links

• J. Fulman, Random matrix theory over finite fields, Bull. Amer. Math. Soc., 39 (No. 1, 2002), 51-85, MR1864086 (2002i:60012). See top of page 70, Eq. 1, with k=1.

Crossrefs

k=0 gives A010815.

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

Programs

Maple

fGL:=proc(k) local a,i,r;
a:=x^(k^2)/mul((1-x^i)^2,i=1..k);
a:=a*mul(1-x^r,r=1..101);
series(a,x,101);
seriestolist(%);
end;fGL(1);

Mathematica

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 *)

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)}
for(n=0, 100, print1(a(n), ", ")) \\ Paul D. Hanna, Dec 14 2015

Formula

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