A136097 - OEIS

%I #10 Jan 10 2023 18:20:23

%S 1,-1,5,-93,6477,-1733677,1816333805,-7526310334829,

%T 124031223014725741,-8152285307423733458541,

%U 2140200604371078953284092525,-2245805993494514875022552272042605,9423041917569791458584837551185555483245

%N a(n) = A135951(n) /[(2^(n+1)-1) * 2^(n*(n-1)/2)].

%C A135951 is the central terms of A135950; A135950 is the matrix inverse of A022166; A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.

%F Conjecture: the n-th central term of the matrix inverse of the triangle of Gaussian binomial coefficients in q is divisible by [(q^(n+1)-1)/(q-1) * q^(n*(n-1)/2)] for n>=0 and integer q > 1.

%F a(n) = (-1)^n * A015030(n) where A015030 is 2-Catalan numbers. - _Michael Somos_, Jan 10 2023

%t Table[(-1)^n QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 16 2016 *)

%o (PARI) a(n)=local(q=2,A=matrix(2*n+1,2*n+1,n,k,if(n>=k,if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1); A[2*n+1,n+1]/( (q^(n+1)-1)/(q-1) * q^(n*(n-1)/2) )

%Y Cf. A135951, A135950, A022166.

%K sign

%O 0,3

%A _Paul D. Hanna_, Dec 13 2007