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 A215406 A ranking algorithm for the lexicographic ordering of the Catalan families. 4
 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 COMMENTS See Antti Karttunen's code in A057117. Karttunen writes: "Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES (Kreher and Stinson) book." For all n>0, a(A014486(n)) = n = A080300(A014486(n)). The sequence A080300 differs from this one in that it gives 0 for those n which are not found in A014486. - Antti Karttunen, Aug 10 2012 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 A. Karttunen, Catalan's Triangle and ranking of Mountain Ranges D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998. F. Ruskey, Algorithmic Solution of Two Combinatorial Problems, Thesis, Department of Applied Physics and Information Science, University of Victoria, 1978. MAPLE A215406 := proc(n) local m, a, y, t, x, u, v; m := iquo(A070939(n), 2); a := A030101(n); y := 0; t := 1; for x from 0 to 2*m-2 do     if irem(a, 2) = 1 then y := y + 1     else u := 2*m - x;          v := m-1 - iquo(x+y, 2);          t := t + A037012(u, v);          y := y - 1 fi;     a := iquo(a, 2) od; A014137(m) - t end: seq(A215406(i), i=0..199); # Peter Luschny, Aug 10 2012 MATHEMATICA A215406[n_] := Module[{m, d, a, y, t, x, u, v}, m = Quotient[Length[d = IntegerDigits[n, 2]], 2]; a = FromDigits[Reverse[d], 2]; y = 0; t = 1; For[x = 0, x <= 2*m - 2, x++, If[Mod[a, 2] == 1, y++, u = 2*m - x; v = m - Quotient[x + y, 2] - 1; t = t - Binomial[u - 1, v - 1] + Binomial[u - 1, v]; y--]; a = Quotient[a, 2]]; (1 - I*Sqrt[3])/2 - 4^(m + 1)*Gamma[m + 3/2]*Hypergeometric2F1[1, m + 3/2, m + 3, 4]/(Sqrt[Pi]*Gamma[m + 3]) - t]; Table[A215406[n] // Simplify, {n, 0, 86}] (* Jean-François Alcover, Jul 25 2013, translated and adapted from Peter Luschny's Maple program *) PROG (Sage) def A215406(n) : # CatalanRankGlobal(n)     m = A070939(n)//2     a = A030101(n)     y = 0; t = 1     for x in (1..2*m-1) :         u = 2*m - x; v = m - (x+y+1)/2         mn = binomial(u, v) - binomial(u, v-1)         t += mn*(1 - a%2)         y -= (-1)^a         a = a//2     return A014137(m) - t CROSSREFS Cf. A213704, A057117, A057164, A057505, A057506, A057501, A057502, A057511, A057512, A057123, A057117, A057509, A057510, A057161, A057162, A072766, A071654, A071652, A075161, A061856, A072635, A072788, A057518, A075168, A080119, A057120, A072773. Sequence in context: A301805 A260236 A122462 * A231717 A253315 A334138 Adjacent sequences:  A215403 A215404 A215405 * A215407 A215408 A215409 KEYWORD nonn,look AUTHOR Peter Luschny, Aug 09 2012 STATUS approved

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Last modified July 30 06:20 EDT 2021. Contains 346348 sequences. (Running on oeis4.)