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A290974 Alternating sum of row 2n of A022166. 0
1, -1, 7, -217, 27559, -14082649, 28827182503, -236123451882073, 7737057147819885991, -1014103817421900276726361, 531681448124675830384033629607, -1115016280616112042365706510363949657, 9353433376690281791373262192784600640357799 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The alternating row sums of A022166(n,k) is zero when n is odd.

LINKS

Table of n, a(n) for n=0..12.

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

Cf. A005329, A022166.

Sequence in context: A114939 A193224 A319538 * A145107 A231488 A231487

Adjacent sequences:  A290971 A290972 A290973 * A290975 A290976 A290977

KEYWORD

sign

AUTHOR

Geoffrey Critzer, Aug 16 2017

STATUS

approved

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Last modified October 2 06:43 EDT 2022. Contains 357191 sequences. (Running on oeis4.)