A226302 a(n) = P_n(-1), where P_n(x) is a certain polynomial arising in the enumeration of tatami mat coverings.
1, -1, 2, -4, 6, -14, 20, -48, 70, -166, 252, -584, 924, -2092, 3432, -7616, 12870, -28102, 48620, -104824, 184756, -394404, 705432, -1494240, 2704156, -5692636, 10400600, -21785872, 40116600, -83688344, 155117520, -322494208, 601080390, -1246068806, 2333606220
Offset: 2
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 2..1000
- A. Erickson and F. Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013
Programs
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Maple
A226302 := proc(n) if type(n,even) then A000984(n/2-1) ; else -A082590((n-3)/2) ; end if; end proc: # R. J. Mathar, Nov 06 2013
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Mathematica
Rest[Rest[CoefficientList[Series[x^2*(1/Sqrt[1-4*x^2] - x/((1-2*x^2)*Sqrt[1-4*x^2])), {x, 0, 30}], x]]] (* Vaclav Kotesovec, Jun 14 2015, after Sergei N. Gladkovskii *) max = 30; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - 2*x^2 - 2*x*(1 - 2*x^2)^2*(2*k+1)/( 2*x*(1 - 2*x^2)*(2*k+1) - (k+1)/(1 - x/g[k+1] )); gf = 1 - x/g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jun 14 2015, after Sergei N. Gladkovskii *) a = DifferenceRoot[Function[{a, n}, {(-(6*n^2) + 2*n + 4)*a[n+2] + (n^2 + n - 2)*a[n+4] + 8*(n - 1)*n*a[n] - 4*n*a[n+1] + 2*n*a[n+3] == 0, a[2] == 1, a[3] == -1, a[4] == 2, a[5] == -4}]]; Table[a[n], {n, 2, 36}] (* Jean-François Alcover, Feb 23 2019 *) -
PARI
Vec(x^2*(1/sqrt(1-4*x^2) - x/((1-2*x^2)*sqrt(1-4*x^2))) + O(x^50)) \\ G. C. Greubel, Jan 29 2017
Formula
Conjecture: (-n+2)*a(n) +(-n+2)*a(n-1) +2*(3*n-11)*a(n-2) +2*(3*n-14)*a(n-3) +4*(-2*n+9)*a(n-4) +8*(-n+6)*a(n-5)=0. - R. J. Mathar, Nov 06 2013
G.f. (for offset 0): 1/sqrt(1-4*x^2) - x/((1-2*x^2)*sqrt(1-4*x^2)) = 1 - x/W(0), where W(k)= 1 - 2*x^2 - 2*x*(1 - 2*x^2)^2*(2*k+1)/( 2*x*(1 - 2*x^2)*(2*k+1) - (k+1)/(1 - x/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jun 13 2015
Recurrence (for n>5): (n-5)*(n-2)*a(n) = -2*(n-4)*a(n-1) + 2*(n-5)*(3*n-10)*a(n-2) + 4*(n-4)*a(n-3) - 8*(n-5)*(n-4)*a(n-4). - Vaclav Kotesovec, Jun 14 2015
a(n) ~ (-1)^n * 2^(n-3/2) / sqrt(Pi*n). - Vaclav Kotesovec, Jun 14 2015
Comments