A022553 -id:A022553 - cp's OEIS frontend

A006177 Witt vector *2!/2!.

1, 1, 3, 8, 25, 72, 245, 772, 2692, 8925, 32065, 109890, 400023, 1402723, 5165327, 18484746, 68635477, 248339122, 930138521, 3406231198, 12810761323, 47306348881, 178987624513, 665627041157, 2528210175630, 9456885664122

Offset: 1

Simon Plouffe

nonn

The Somos transform sends sequence {a(n)} to sequence with g.f. Product_{i=1..n} 1/(1-a(i)*x^i).

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Cf. A006173, A006973.

Inverse Somos transform of A000108. - Wouter Meeussen, Aug 20 2002

Witt transform of A022553.

Edited by Christian G. Bower, Aug 20 2002, Aug 28 2002

A178738 Moebius inversion of a sequence related to powers of 2.

Original entry on oeis.org

1, -1, -1, 1, 2, -3, -5, 9, 15, -27, -49, 89, 164, -304, -565, 1057, 1987, -3745, -7085, 13445, 25575, -48771, -93210, 178481, 342392, -657935, -1266205, 2440323, 4709403, -9099507, -17602325, 34087058, 66076421, -128207979, -248983641

Offset: 1

F. Chapoton, Jun 08 2010

Only odd indices make sense. The given sequence is a(1), a(3), a(5), etc.
This should be related to the Coxeter transformations for the posets of diagonally symmetric paths in an n*n grid. - F. Chapoton, Jun 11 2010
Start from 1, 1, -2, -2, -4, -4, 8, 8, 16, 16, -32, -32, -64, -64, 128, ... which is A016116(n-1) with negative signs in blocks of 4, assuming offset 1. The Mobius transform of that sequence is b(n) = 1, 0, -3, -3, -5, -2, 7, 10, 18, 20, -33, -25, -65, -72, 135, 120, ... for n >= 1, and the current sequence is a(n) = b(2n-1)/(2n-1). - R. J. Mathar, Oct 29 2011

			b(1)=1*1; b(3)=-1*3; ...; b(9)=2*9.

Similar to A022553 and A131868

Also related to A178749. - F. Chapoton, Jun 11 2010

A := proc(n)
        (-1)^binomial(floor((n+1)/2),2) * 2^floor((n-1)/2) ;
end proc:
L := [seq(A(n),n=1..40)] ;
b := MOBIUS(L) ;
for i from 1 to nops(b) by 2 do
        printf("%d,", op(i,b)/i) ;
end do: # R. J. Mathar, Oct 29 2011

b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;
a[n_] := b[2n - 1];
a /@ Range[35] (* Jean-François Alcover, Mar 16 2020 *)

def suite(n):
    return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n
[suite(n) for n in range(1,22,2)]

I would like a more precise definition. - N. J. A. Sloane, Jun 08 2010

A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

Original entry on oeis.org

1, -1, -1, 1, 1, -1, -3, 4, 8, -13, -23, 39, 71, -121, -229, 400, 757, -1354, -2559, 4625, 8799, -16021, -30671, 56316, 108166, -200047, -385210, 716429, 1383331, -2585173, -5003791, 9391680, 18214565, -34318117, -66674463, 126044208, 245273927, -465067981

Offset: 1

F. Chapoton, Jun 09 2010

sign

This should be related to the Coxeter transformation for the Tamari lattices.

The source sequence is 1, -1, -2, 3, 6, -10, -20, 35, 70, -126, ... (A001405). Its Mobius transform is 1, -2, -3, 4, 5, -6, -21, 32, 72, -130, -253, 468, 923, ... and division of each term through n generates a(n). - R. J. Mathar, Jul 23 2012

			G.f. = x - x^2 - x^3 + x^4 + x^5 - x^6 - 3*x^7 + 4*x^8 + 8*x^9 - 13*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
F. Chapoton, Le module dendriforme sur le groupe cyclique, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2333-2350. In French.

Similar to A022553, A131868 and A178738.

Also related to A163210.

with(numtheory):
a:= n-> add(mobius(n/d)*[1$2, -1$2][1+irem(d, 4)]*
        binomial(d-1, iquo(d-1, 2)), d=divisors(n))/n:
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 05 2013

a[ n_] := If[ n < 1, 0, DivisorSum[ n, MoebiusMu[ n/#] (-1)^Quotient[ #, 2] Binomial[ # - 1, Quotient[ # - 1, 2]] &] / n]; (* Michael Somos, Sep 14 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, moebius(n/d) * (-1)^(d\2) * binomial(d-1, (d-1)\2)) / n)}; /* Michael Somos, Dec 23 2014 */

def lam(n):
    return (-1)**binomial(n, 2) * binomial(n - 1, (n - 1) // 2)
def a(n):
    return sum(moebius(n // d) * lam(d) for d in divisors(n)) // n
[a(n) for n in range(1, 20)]

A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 1, 3, 11, 35, 118, 392, 1336, 4587, 15986, 56231, 199854, 716014, 2584742, 9390656, 34315811, 126039218, 465062362, 1723066193, 6407806833, 23910159818, 89493721076, 335912335304, 1264105728831, 4768446686910, 18027215660947, 68291877609003

Offset: 0

Tilman Piesk, Aug 07 2025

A022553(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A000048 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 2, 6, 22, 70, 236, 784, 2672, 9174, 31972, 112462, 399708, 1432028, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |   this       A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A022553(n) A000048(n) | 2*a(n)    a(n)
  0 |         1          1  |     0       0
  1 |         1          1  |     0       0
  2 |         1          1  |     0       0
  3 |         3          1  |     2       1
  4 |         8          2  |     6       3
  5 |        25          3  |    22      11
  6 |        75          5  |    70      35
  7 |       245          9  |   236     118
  8 |       800         16  |   784     392
  9 |      2700         28  |  2672    1336
 10 |      9225         51  |  9174    4587
Examples for n=5 with necklaces of length 10:
The total number of necklaces is A003239(5) = 26.
Only A386946(5) = 1 of them is periodic, namely 0101010101.
The other A022553(5) = 25 are primitive.
A000048(5) = 3 among those are self-complementary:
 0000011111
 0001011101
 0010011011
The remaining 22 necklaces form a(5) = 11 complement pairs:
 0000101111 0000111101
 0000110111 0001111001
 0000111011 0001001111
 0001010111 0001110101
 0001011011 0010011101
 0001100111 0001110011
 0001101011 0010100111
 0001101101 0010010111
 0010101011 0011010101
 0010101101 0010110101
 0010110011 0011001101

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = (A022553(n) - A000048(n)) / 2.

A385666 Triangle read by rows: T(n,k) is the number of 2n-bead balanced binary necklaces with period length 2n/k.

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 8, 1, 0, 1, 25, 0, 0, 0, 1, 75, 3, 1, 0, 0, 1, 245, 0, 0, 0, 0, 0, 1, 800, 8, 0, 1, 0, 0, 0, 1, 2700, 0, 3, 0, 0, 0, 0, 0, 1, 9225, 25, 0, 0, 1, 0, 0, 0, 0, 1, 32065, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 112632, 75, 8, 3, 0, 1, 0, 0, 0, 0, 0, 1, 400023

Offset: 1

Tilman Piesk, Jul 16 2025

There are A003239(n) balanced binary necklaces of length 2n. (Central numbers of A047996.)
T(n,k) is the number of those that can be rotated into themselves in k different ways (at least 1 for the trivial rotation).
A022553(n) necklaces (corresponding to Lyndon words) have only the trivial rotation.
All columns have the same positive entries, each preceded by k-1 zeros.
Compare triangle A385665, which counts only self-complementary balanced binary necklaces.

			Triangle begins:
         k     1   2  3  4  5  6  7  8  9 10 11 12 13 14 15 16    A003239(n)
  n
  1            1   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .            1
  2            1   1  .  .  .  .  .  .  .  .  .  .  .  .  .  .            2
  3            3   .  1  .  .  .  .  .  .  .  .  .  .  .  .  .            4
  4            8   1  .  1  .  .  .  .  .  .  .  .  .  .  .  .           10
  5           25   .  .  .  1  .  .  .  .  .  .  .  .  .  .  .           26
  6           75   3  1  .  .  1  .  .  .  .  .  .  .  .  .  .           80
  7          245   .  .  .  .  .  1  .  .  .  .  .  .  .  .  .          246
  8          800   8  .  1  .  .  .  1  .  .  .  .  .  .  .  .          810
  9         2700   .  3  .  .  .  .  .  1  .  .  .  .  .  .  .         2704
 10         9225  25  .  .  1  .  .  .  .  1  .  .  .  .  .  .         9252
 11        32065   .  .  .  .  .  .  .  .  .  1  .  .  .  .  .        32066
 12       112632  75  8  3  .  1  .  .  .  .  .  1  .  .  .  .       112720
 13       400023   .  .  .  .  .  .  .  .  .  .  .  1  .  .  .       400024
 14      1432613 245  .  .  .  .  1  .  .  .  .  .  .  1  .  .      1432860
 15      5170575   . 25  .  3  .  .  .  .  .  .  .  .  .  1  .      5170604
 16     18783360 800  .  8  .  .  .  1  .  .  .  .  .  .  .  1     18784170
Examples for n=4 with necklaces of length 8:
T(4, 1) = 8 necklaces have k=1 rotation, i.e. rotating 0 places:
 00001111, 00010111, 00011011, 00011101, 00100111, 00101011, 00101101, 00110101
T(4, 2) = 1 necklace has k=2 rotations:
 00110011 can be rotated onto itself by rotating 0 or 4 places.
T(4, 4) = 1 necklace has k=4 rotations:
 01010101 can be rotated onto itself by rotating 0, 2, 4 or 6 places.

Tilman Piesk, Rows 1..32, flattened
Tilman Piesk, Triangle T(n,k)*2*n/k with row sums 2n choose n
Tilman Piesk, List of imprimitive necklaces for n=1...15
Tilman Piesk, List of primitive necklaces for n=1...8

Cf. A022553, A003239, A385665.

T(n,k) = A022553(n/k) iff n divisible by k, otherwise 0.

A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 5, 1, 10, 4, 27, 1, 88, 1, 247, 29, 810, 1, 2780, 1, 9260, 249, 32067, 1, 113520, 26, 400025, 2704, 1432868, 1, 5179905, 1, 18784170, 32069, 68635479, 271, 252201136, 1, 930138523, 400027, 3446168660, 1, 12817096533, 1, 47820447036, 5173304

Offset: 0

Tilman Piesk, Aug 10 2025

A003239(n) is the number of 2n-bead balanced binary necklaces. A022553(n) among them are primitive.
The remaining a(n) necklaces are periodic.
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904     A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553      this    | A003239 |
                     +-----------------------+---------+

			  n | A003239(n) A022553(n) | a(n)
  0 |         1          1  |   0
  1 |         1          1  |   0
  2 |         2          1  |   1
  3 |         4          3  |   1
  4 |        10          8  |   2
  5 |        26         25  |   1
  6 |        80         75  |   5
  7 |       246        245  |   1
  8 |       810        800  |  10
  9 |      2704       2700  |   4
 10 |      9252       9225  |  27
 11 |     32066      32065  |   1
 12 |    112720     112632  |  88
 13 |    400024     400023  |   1
 14 |   1432860    1432613  | 247
 15 |   5170604    5170575  |  29
 16 |  18784170   18783360  | 810
There are A003239(8) = 810 balanced binary necklaces of length 16. A022553(8) = 800 of them are primitive. a(n) = 10 are not. See A387130 for a list.

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = A003239(n) - A022553(n).

a(n) = A115118(n) + 2 * A387130(n).

A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 11, 0, 39, 0, 118, 12, 395, 0, 1372, 0, 4601, 119, 15986, 0, 56662, 11, 199854, 1337, 716135, 0, 2589376, 0, 9391051, 15987, 34315811, 129, 126096824, 0, 465062362, 199855, 1723071186, 0, 6408523001, 0, 23910175807, 2586090, 89493721076

Offset: 0

Tilman Piesk, Aug 17 2025

A386946(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A115118 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 0, 0, 0, 2, 0, 6, 2, 22, 0, 78, 0, 236, 24, 790, 0, 2744, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904      this    | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A386946(n) A115118(n) | 2*a(n)    a(n) | A386388(n) A383904(n)
  0 |         0          0  |     0       0  |         0          0
  1 |         0          0  |     0       0  |         0          0
  2 |         1          1  |     0       0  |         0          0
  3 |         1          1  |     0       0  |         1          1
  4 |         2          2  |     0       0  |         3          3
  5 |         1          1  |     0       0  |        11         11
  6 |         5          3  |     2       1  |        36         35
  7 |         1          1  |     0       0  |       118        118
  8 |        10          4  |     6       3  |       395        392
  9 |         4          2  |     2       1  |      1337       1336
 10 |        27          5  |    22      11  |      4598       4587
 11 |         1          1  |     0       0  |     15986      15986
 12 |        88         10  |    78      39  |     56270      56231
 13 |         1          1  |     0       0  |    199854     199854
 14 |       247         11  |   236     118  |    716132     716014
 15 |        29          5  |    24      12  |   2584754    2584742
 16 |       810         20  |   790     395  |   9391051    9390656
Examples for n=8 with necklaces of length 16:
The total number of necklaces is A003239(8) = 810.
A022553(8) = 800 of them are primitive.
The other A386946(8) = 10 are periodic.
A115118(8) = 4 among those are self-complementary:
 0000111100001111
 0010110100101101
 0011001100110011
 0101010101010101
The remaining 6 necklaces form a(8) = 3 complement pairs:
 0001011100010111 0001110100011101
 0001101100011011 0010011100100111
 0010101100101011 0011010100110101

Tilman Piesk, Table of n, a(n) for n = 0..1000

Cf. A386946, A115118, A386388, A383904.

a(n) = (A386946(n) - A115118(n)) / 2.

a(n) = A386388(n) - A383904(n).

A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

Original entry on oeis.org

0, 0, 0, 0, 0, 8, 32, 168, 616, 2380, 8464, 30760, 109612, 394816, 1420616, 5149940, 18736128, 68553728, 251899620, 929814984, 3445425136, 12814382452, 47817520376, 178982546512, 671813585080, 2528191984496, 9536849432000

Offset: 0

David W. Wilson

nonn

The number of length 2n balanced binary Lyndon words is A022553(n) and the number which are equivalent to their reverse, complement and reversed complement are respectively A045680(n), A000048(n) and A000740(n). - Andrew Howroyd, Sep 29 2017

Jean-François Alcover, Table of n, a(n) for n = 0..100
Index entries for sequences related to Lyndon words

Cf. A000048, A000740, A022553, A045675, A045680, A045683.

a22553[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n)];
a45680[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]] &]];
a48[n_] := If[n == 0, 1, Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a740[n_] := Sum[MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}];
b[n_] := Module[{t = 0, r = n}, If[n == 0, 1, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]]; t + 2^Quotient[r, 2]];
a45683[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#]*b[#] &]];
a[n_] := If[n == 0, 0, a22553[n] - a45680[n] - a48[n] - a740[n] + 2 a45683[n]];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 23 2019 *)

From Andrew Howroyd, Sep 28 2017: (Start)
Moebius transform of A045675.
a(n) = A022553(n) - A045680(n) - A000048(n) - A000740(n) + 2*A045683(n).
(End)

A074655 Number of Lyndon words (aperiodic necklaces) with 3n beads of 3 colors, n beads of each color.

Original entry on oeis.org

1, 2, 14, 186, 2880, 50450, 952854, 19003474, 394394880, 8439756660, 185033201150, 4137181680698, 94020326259264, 2166105078791446, 50489825369325118, 1188777328563863850, 28236363841594782720, 675879582290807439794, 16289254212695836475436

Offset: 0

Christian G. Bower, Aug 29 2002

nonn

Index entries for sequences related to Lyndon words

Cf. A029808, A074651, A022553 (2n of 2 colors), A074656 (4n of 4 colors).

a(n) = 1/(3n) * Sum_{d|n} mu(n/d) * (3d)! / d!^3, a(0) = 1.

a(n) = A029808(n)*2 = A074651(n)/3.

a(0)=1 prepended by Alois P. Heinz, Aug 24 2015

A086655 (binomial(2p,p)-2)/(2p) where p runs through the primes.

Original entry on oeis.org

1, 3, 25, 245, 32065, 400023, 68635477, 930138521, 178987624513, 518401146543811, 7506908923471953, 23596358977508462295, 5180299766448679532059, 77180849825857621779893, 17294692982395428197325697, 59174764703889164618206676203, 206412709386903552066428869508701

Offset: 1

Benoit Cloitre, Sep 11 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..263

Cf. A000984, A022553.

Table[(Binomial[2*p, p] - 2)/(2*p), {p, Prime[Range[18]]}] (* Amiram Eldar, Apr 21 2025 *)

a(n) = A022553(prime(n)).

Typo in definition corrected by Vladimir Shevelev, Sep 05 2010

cp's OEIS Frontend

A006177 Witt vector *2!/2!.

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A178738 Moebius inversion of a sequence related to powers of 2.

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A178749 n*a(n) provides the Moebius transform of signed central binomial coefficients.

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A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

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A385666 Triangle read by rows: T(n,k) is the number of 2n-bead balanced binary necklaces with period length 2n/k.

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A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

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A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces.

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A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

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A086655 (binomial(2p,p)-2)/(2p) where p runs through the primes.