A075161 -id:A075161 - cp's OEIS frontend

A106191 Expansion of sqrt(1-4x)/(1-x).

1, -1, -3, -7, -17, -45, -129, -393, -1251, -4111, -13835, -47427, -164999, -581023, -2066823, -7415703, -26805393, -97520733, -356810313, -1312087713, -4846614093, -17974854933, -66907388973, -249872516253, -935991743553, -3515800038201, -13239692841105

Offset: 0

Paul Barry, Apr 24 2005

Row sums of number triangle A106190. Partial sums of A002420.

For n >= 1, the absolute values also give the iterates of A122237, starting from 0. (A122237(0), A122237(A122237(0)), A122237(A122237(A122237(0))), ...), this stems from the fact that the sequence gives the positions of terms with binary expansion 1(10){n-1}0 in A014486 (see A080675).

|a(n)| = A080300(A080675(n)) = A075161(A001348(n)) (for n >= 1) = A075163(A000244(A008578(n-2))) = A014137(n-1)+A014138(n-2) = 2*A014137(n-1)-1, for n >= 2 (because binomial(2n+2, n+1)/(2n+1) = 2*A000108(n)).

a(n) = Sum_{k=0..n} binomial(2k, k)/(1-2k).
G.f.: (2/(1-x))/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
D-finite with recurrence: a(0)=1, a(1)=-1; for n>1, a(n) = (1/n)*((5*n-6)*a(n-1) - (4*n-6)*a(n-2)). - Tani Akinari, Aug 25 2013

Barry's formula made more succinct, as well as comments regarding interpretation as absolute values added by Antti Karttunen, Sep 14 2006

A075163 Position of A075165(n) in A014486 plus one.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 8, 7, 11, 24, 13, 66, 25, 12, 9, 198, 16, 627, 27, 26, 67, 2057, 14, 15, 199, 17, 69, 6919, 30, 23715, 18, 68, 628, 29, 41, 82501, 2058, 200, 28, 290513, 72, 1033413, 201, 31, 6920, 3707853, 32, 38, 39, 629, 630, 13402698, 44, 71, 70, 2059

Offset: 1

Antti Karttunen, Sep 13 2002

nonn

See A075166.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse of A075164. a(n) = A075161(n-1)+1.

a(n) = A106453(A106442(n)). - Antti Karttunen, May 09 2005

A075162 Position of A014486(n) in A075165, minus one.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 7, 15, 6, 9, 14, 11, 23, 24, 17, 26, 31, 63, 80, 511, 255, 65535, 10, 13, 20, 19, 39, 34, 29, 44, 47, 95, 134, 767, 383, 98303, 48, 49, 74, 35, 71, 124, 53, 242, 127, 1023, 728, 32767, 4095, 16777215, 624, 161, 19682, 33554431, 262143, 6560

Offset: 0

Antti Karttunen, Sep 13 2002

nonn

See A075166.

A. Karttunen, Alternative Catalan Orderings (with the complete Scheme source)
Index entries for sequences that are permutations of the natural numbers

Inverse of A075161. a(n) = A075164(n+1)-1. Cf. A075157, A075169.

A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Aug 09 2012

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

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.

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.

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

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 *)

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

A106451 Position of A106455(n+1) in A014486.

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 11, 15, 9, 12, 23, 10, 16, 8, 19, 29, 65, 40, 14, 24, 67, 13, 197, 66, 30, 26, 25, 43, 626, 20, 2058, 52, 70, 82, 2056, 198, 72, 41, 6918, 38, 628, 68, 33, 203, 23714, 34, 28, 627, 53, 200, 199, 85, 82500, 27, 204, 71, 290512, 124, 1033412

Offset: 0

Antti Karttunen, May 09 2005

nonn

See A106456.

Inverse: A106452. a(n) = A106453(n+1)-1. GF(2)[X]-analog of A075161.

cp's OEIS Frontend

A106191 Expansion of sqrt(1-4x)/(1-x).

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A075163 Position of A075165(n) in A014486 plus one.

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A075162 Position of A014486(n) in A075165, minus one.

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A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

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Sage

A106451 Position of A106455(n+1) in A014486.

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