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A290974
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Alternating sum of row 2n of A022166.
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0
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1, -1, 7, -217, 27559, -14082649, 28827182503, -236123451882073, 7737057147819885991, -1014103817421900276726361, 531681448124675830384033629607, -1115016280616112042365706510363949657, 9353433376690281791373262192784600640357799
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OFFSET
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0,3
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COMMENTS
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The alternating row sums of A022166(n,k) is zero when n is odd.
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LINKS
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FORMULA
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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
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MATHEMATICA
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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];
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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