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A121703
Number of alternating separable permutations.
1
1, 2, 4, 8, 20, 48, 132, 344, 996, 2720, 8132, 22888, 69940, 201040, 624132, 1822136, 5725124, 16915008, 53648772, 160012232, 511360340, 1536928624, 4942300804, 14949122328, 48322714020, 146946942688, 477105960772, 1457491035944, 4750171491956, 14568377075344
OFFSET
1,2
COMMENTS
The separable permutations are those avoiding 2413 and 3142 and are counted by the large Schroeder numbers (A006318). The alternating permutations are counted by the Euler numbers (A000111).
LINKS
Robert Brignall, Sophie Huczynska, and Vincent Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.
FORMULA
G.f. satisfies f^3-(2x^2-5x+4)f^2-(4x^3+x^2-8x)f-(2x^4+5x^3+4x^2)=0.
EXAMPLE
a(4)=8 because of the 10 alternating permutations of length 4, 2413 and 3142 are not separable.
MATHEMATICA
nmax = 40; aa = ConstantArray[0, nmax]; aa[[1]] = 1; aa[[2]] = 2; Do[f = Sum[aa[[k]]*x^k, {k, 1, j - 1}] + koef*x^j; sol = Solve[SeriesCoefficient[f^3 - (2*x^2 - 5*x + 4)*f^2 - (4*x^3 + x^2 - 8*x)*f - (2*x^4 + 5*x^3 + 4*x^2), {x, 0, j + 2}] == 0, koef][[1]]; aa[[j]] = koef /. sol[[1]], {j, 3, nmax}]; aa (* Vaclav Kotesovec, Jul 07 2024 *)
CROSSREFS
Cf. A121704.
Sequence in context: A056952 A369496 A225585 * A301467 A358056 A275070
KEYWORD
nonn
AUTHOR
Vincent Vatter, Aug 16 2006
EXTENSIONS
a(13) and beyond corrected by Vaclav Kotesovec, Jul 07 2024
STATUS
approved