OFFSET
0,3
COMMENTS
The alternating row sums of A022166(n,k) is zero when n is odd.
LINKS
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
A. Nijenhuis, A. E. Solow, and H. S. Wilf, Bijective Methods in the Theory of Finite Vector Spaces, Journal of Combinatorial Theory, Series A 37,(1984), 80-84.
FORMULA
a(n) = Sum_{k=0..2n} (-1)^k A022166(2n,k).
a(0) = 1, a(n) = (1 - 2^(2n-1))*a(n-1).
a(n)/A005329(2n) is the coefficient of z^(2n) in the expansion of eq(-z)*eq(z) where eq(z) is the q-exponential function.
O.g.f.: Sum_{n>=0} a(n)*x^n = 1/(1 + (q-1)*x/(1 + q*(q^2-1)*x/(1 + q^2*(q^3-1)*x/(1 + q^3*(q^4-1)*x/(1 + q^4*(q^5-1)*x/(1 + q^5*(q^6-1)*x/(1 + ...))))))), a continued fraction, when evaluated at q = 2. - Paul D. Hanna, Aug 29 2020
O.g.f.: Sum_{n>=0} a(n)*x^(2*n) = Sum_{n>=0} (-x)^k / Product{k=0..n} (1 - 2^k*x). - Paul D. Hanna, Aug 29 2020
MATHEMATICA
nn = 26; eq[z_] :=Sum[z^n/FunctionExpand[QFactorial[n, q]], {n, 0, nn}]; Select[Table[FunctionExpand[QFactorial[n, q]] /. q -> 2, {n, 0, nn}] CoefficientList[Series[eq[-z]*eq[z] /. q -> 2, {z, 0, nn}], z], # != 0 &]
a[n_Integer] := a[n] = 2 QPochhammer[1/2, 4, n + 1];
Table[a[n], {n, 0, 12}] (* Vladimir Reshetnikov, Sep 23 2021 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Geoffrey Critzer, Aug 16 2017
STATUS
approved