A007727 -id:A007727 - cp's OEIS frontend

A005354 Number of asymmetric planar trees with n nodes.

1, 1, 0, 0, 0, 1, 3, 9, 28, 85, 262, 827, 2651, 8626, 28507, 95393, 322938, 1104525, 3812367, 13266366, 46504495, 164098390, 582521687, 2079133141, 7457788295, 26872946466, 97238824018, 353218128299, 1287657977946, 4709784136316

Offset: 0

N. J. A. Sloane, Simon Plouffe, Susanna Cuyler

a(13) in the Labelle table is a typographical error. - R. J. Mathar, Feb 03 2010

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Gilbert Labelle, Counting asymmetric enriched trees, J. Symbolic Comput. 14 (1992), no. 2-3, 211-242.
Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
T. Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.
Index entries for sequences related to trees

Cf. A000108, A002995, A022553.

From R. J. Mathar, Feb 03 2010: (Start)
A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A007727 := proc(n) local a,d; a := 0 ; for d in numtheory[divisors](n) do a := a+binomial(2*d,d)*numtheory[mobius](n/d) ; end do ; a ; end proc;
A022553 := proc(n) A007727(n)/2/n ; end proc:
A005354 := proc(n) local a; if n <=1 then 1; else a := A022553(n-1) ; a := a-A000108(n-1)/2 ; if type(n,'even') then a := a-A000108(n/2-1)/2 ; end if; a ; end if; end proc: seq(A005354(n),n=0..20) ; (End)

a[0] = a[1] = 1; a[n_] := DivisorSum[n-1, MoebiusMu[(n-1)/#]*Binomial[2#, #]&]/(2(n-1)) - CatalanNumber[n-1]/2 - Boole[EvenQ[n]]*CatalanNumber[n/2 - 1]/2; Table[a[n], {n, 0, 29}] (* Jean-François Alcover, May 09 2012, after R. J. Mathar, updated Jan 31 2018 *)

From Christian G. Bower, Dec 15 1999: (Start)
G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A022553(n-1) and C is g.f. of A000108(n-1).
a(n) = A022553(n-1) - A000108(n-2)/2 - (if n is even) A000108(n/2-1)/2. (End)

More terms from Christian G. Bower, Dec 15 1999

A045666 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally inequivalent to reverse, complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 80, 384, 2352, 9856, 42840, 169280, 676720, 2630688, 10265216, 39777248, 154498200, 599556096, 2330826752, 9068386320, 35332969392, 137817005440, 538204062984, 2103970896544, 8233197139552, 32247052083840

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A007727, A045662, A045663, A045664, A045665, A045684.

a(n) = 2*n*A045684(n).

a(n) = A007727(n) - A045662(n) - A045663(n) - A045664(n) + 2*A045665(n). - Andrew Howroyd, Sep 14 2019

A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 2, 0, 3, 3, 0, 4, 4, 4, 0, 5, 10, 10, 5, 0, 6, 12, 18, 12, 6, 0, 7, 21, 35, 35, 21, 7, 0, 8, 24, 56, 64, 56, 24, 8, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 0, 12, 60

Offset: 1

N. J. A. Sloane, Mar 24 2000

Triangle of number of primitive words over {0,1} of length n that contain k 1's, for n,k >= 1. - Benoit Cloitre, Jun 08 2004

			Triangle begins
  1;
  2,  0;
  3,  3,  0;
  4,  4,  4,  0;
  5, 10, 10,  5,  0;
  6, 12, 18, 12,  6,  0;
  ...

J.-P. Allouche and J. Shallit, Automatic sequences, Cambridge University Press, 2003, p. 29.

Index entries for triangles and arrays related to Pascal's triangle

Cf. A042979, A042980. T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

Same triangle as A050186 except this one does not include column 0.

T[n_, k_] := DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#] &]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)

T(n,k)=sumdiv(gcd(k,n),d,moebius(d)*binomial(n/d,k/d)) \\ Benoit Cloitre, Jun 08 2004

A062791 Moebius transform of A001405 (binomial(n, floor(n/2))).

Original entry on oeis.org

1, 1, 2, 4, 9, 16, 34, 64, 123, 241, 461, 900, 1715, 3396, 6423, 12800, 24309, 48477, 92377, 184500, 352679, 704969, 1352077, 2703168, 5200290, 10398883, 20058174, 40113164, 77558759, 155110827, 300540194, 601067520, 1166802646, 2333581909, 4537567606

Offset: 1

Labos Elemer, Jul 19 2001

nonn

			For n = 7, binomial(7,3) = 35, A001405(7/d) = {binomial(7,3), binomial(1,0)} = {35, 1}, mu(d) = {1, -1}, the sum is a(7) = 35 - 1 = 34.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A001405, A007727, A062798.

with(numtheory):
a:= n-> add(binomial(d, iquo(d, 2))*mobius(n/d), d=divisors(n)):
seq(a(n), n=1..40);  # Alois P. Heinz, Jul 09 2017

a[n_] := DivisorSum[n, Binomial[#, Floor[#/2]] * MoebiusMu[n/#] &]; Array[a, 35] (* Amiram Eldar, May 28 2025 *)

a(n) = sumdiv(n, d, binomial(d, d\2) * moebius(n/d)); \\ Amiram Eldar, May 28 2025

a(n) = Sum_{d|n} A001405(n/d)*mu(d).

Offset corrected by Eric Rowland, Jul 09 2017

cp's OEIS Frontend

A005354 Number of asymmetric planar trees with n nodes.

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A045666 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally inequivalent to reverse, complement and reversed complement.

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A053727 Triangle T(n,k) = Sum_{d|gcd(n,k)} mu(d)*C(n/d,k/d) (n >= 1, 1 <= k <= n).

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A062791 Moebius transform of A001405 (binomial(n, floor(n/2))).

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