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A290974 Alternating sum of row 2n of A022166.

%I A290974 #29 Sep 24 2021 03:23:56
%S A290974 1,-1,7,-217,27559,-14082649,28827182503,-236123451882073,
%T A290974 7737057147819885991,-1014103817421900276726361,
%U A290974 531681448124675830384033629607,-1115016280616112042365706510363949657,9353433376690281791373262192784600640357799
%N A290974 Alternating sum of row 2n of A022166.
%C A290974 The alternating row sums of A022166(n,k) is zero when n is odd.
%H A290974 Geoffrey Critzer, <a href="https://esirc.emporia.edu/handle/123456789/3595">Combinatorics of Vector Spaces over Finite Fields</a>, Master's thesis, Emporia State University, 2018.
%H A290974 A. Nijenhuis, A. E. Solow, and H. S. Wilf, <a href="https://doi.org/10.1016/0097-3165(84)90023-2">Bijective Methods in the Theory of Finite Vector Spaces</a>, Journal of Combinatorial Theory, Series A 37,(1984), 80-84.
%F A290974 a(n) = Sum_{k=0..2n} (-1)^k A022166(2n,k).
%F A290974 a(0) = 1, a(n) = (1 - 2^(2n-1))*a(n-1).
%F A290974 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.
%F A290974 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
%F A290974 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
%t A290974 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 &]
%t A290974 a[n_Integer] := a[n] = 2 QPochhammer[1/2, 4, n + 1];
%t A290974 Table[a[n], {n, 0, 12}] (* _Vladimir Reshetnikov_, Sep 23 2021 *)
%Y A290974 Cf. A005329, A022166.
%K A290974 sign
%O A290974 0,3
%A A290974 _Geoffrey Critzer_, Aug 16 2017