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 A054872 Number of (12345, 13245, 21345, 23145, 31245, 32145)-avoiding permutations. 5
 1, 1, 2, 6, 24, 114, 600, 3372, 19824, 120426, 749976, 4762644, 30723792, 200778612, 1326360048, 8842981848, 59425117152, 402092408346, 2737156004376, 18732169337604, 128806616999184, 889479590046108, 6165939982059600, 42891532191557736, 299307319060137504 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Hankel transform is A083667, the number of different antisymmetric relations on n labeled points. - Paul Barry, Jun 26 2008 Conjectured to be the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {5>1, 1>2, 1>3, 1>4} of length 5. That is, conjectured to be the number of length n+1 permutations having no subsequences of length 5 in which the fifth element is the largest and the first element is the next largest - Sergey Kitaev, Dec 13 2020 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..200 from Vincenzo Librandi) Elena Barcucci, Alberto Del Lungo, Elisa Pergola, Renzo Pinzani, Permutations avoiding an increasing number of length-increasing forbidden subsequences, Discrete Mathematics and Theoretical Computer Science 4, 2000, 31-44. Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019. Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26. Eric Weisstein's MathWorld, Legendre Polynomial. FORMULA G.f.: 1 + x*(2 - 2*x - (1 - 8*x + 4*x^2)^(1/2)). - corrected by Vaclav Kotesovec, Oct 11 2012 a(n) = 2*A047891(n-1), n>=2. - Philippe Deléham, Aug 17 2007 Recurrence: (n-1)*a(n) = 4*(2*n-5)*a(n-1) - 4*(n-4)*a(n-2). - Vaclav Kotesovec, Oct 11 2012 a(n) ~ sqrt(26*sqrt(3)-45)*(4+2*sqrt(3))^n/(sqrt(8*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 11 2012 From Vladimir Reshetnikov, Nov 01 2015: (Start) a(n) = 2^(n-1)*(LegendreP_{n-1}(2) - LegendreP_{n-3}(2))/(2*n-3). For n > 2, a(n) = 6*hypergeom([2-n,3-n], , 3). (End) G.f. satisfies: A(x) = x * Sum_{n>=0} ( A(x)/x + 4*x + x/A(x) )^n / (2*4^n). - Paul D. Hanna, Mar 24 2016 G.f. satisfies: A(x) = x * Sum_{n>=0} ( A(x)/x + 4*x - x/A(x) )^n / 4^n. - Paul D. Hanna, Mar 24 2016 EXAMPLE G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 114*x^5 + 600*x^6 + 3372*x^7 + 19824*x^8 + ... MAPLE Set j=3 in the following: f := (x, j)->1-(j+1)*x- sqrt(1-2*(j+1)*x+(j-1)^2*x^2); t := (x, j)->sum(k!*x^k, k=1..(j-1)); s := (x, j)->x^(j-2)*(j-1)!*(f(x, j))/(2)+ t(x, j); MATHEMATICA Table[SeriesCoefficient[x*(2-2*x-(1-8*x+4*x^2)^(1/2)), {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 11 2012 *) Table[2^(n-1) (LegendreP[n-1, 2] - LegendreP[n-3, 2])/(2n-3), {n, 1, 20}] (* Vladimir Reshetnikov, Nov 01 2015 *) PROG (PARI) x='x+O('x^50); Vec(x*(2-2*x-(1-8*x+4*x^2)^(1/2))) \\ Altug Alkan, Nov 02 2015 CROSSREFS Cf. A000108, A047891. Sequence in context: A245233 A228907 A209625 * A134664 A324133 A171448 Adjacent sequences:  A054869 A054870 A054871 * A054873 A054874 A054875 KEYWORD nonn AUTHOR Elisa Pergola (elisa(AT)dsi.unifi.it), May 26 2000 EXTENSIONS a(0)=1 prepended by Alois P. Heinz, Dec 13 2020 STATUS approved

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Last modified June 25 08:27 EDT 2021. Contains 345453 sequences. (Running on oeis4.)