A066318 -id:A066318 - cp's OEIS frontend

A101925 a(n) = A005187(n) + 1.

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

Offset: 0

Ralf Stephan, Dec 28 2004

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Bisection of A089279. First differences are in A001511.
Cf. A000120, A005187, A046699.
Cf. A032184, A052278, A060055, A066318, A088229, A101926.

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PARI
```
a(n)=1+sum(k=1, n, valuation(k,2)+1)
```

a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))

a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022

def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

A032184 a(n) = 2^n*(n-1)! for n > 1, a(1) = 1.

Original entry on oeis.org

1, 4, 16, 96, 768, 7680, 92160, 1290240, 20643840, 371589120, 7431782400, 163499212800, 3923981107200, 102023508787200, 2856658246041600, 85699747381248000, 2742391916199936000, 93241325150797824000, 3356687705428721664000, 127554132806291423232000

Offset: 1

Christian G. Bower

Previous name was: "CIJ" (necklace, indistinct, labeled) transform of 1, 3, 5, 7, ...

C. G. Bower, Transforms (2).
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 565.
Index entries for sequences related to necklaces.

Apart from the initial term, same as A066318.

A032184:=n->if n>1 then 2^n*(n-1)! else 1 fi: seq(A032184(n), n=1..30); # Wesley Ivan Hurt and M. F. Hasler, Jan 15 2017

Join[{1},Table[2^n (n-1)!,{n,2,20}]] (* Harvey P. Dale, Oct 08 2017 *)

apply( A032184=n->(n-1)!<M. F. Hasler, Jan 15 2017

a(n) = 2^n*(n-1)! for n > 1, a(1) = 1.
E.g.f.: (1 + 2*x)/(1 - 2*x). - Paul Barry, May 26 2003 [This e.g.f. yields the sequence (a(n+1): n >= 0). - M. F. Hasler, Jan 15 2017]
a(n) + 2*(-n+1)*a(n-1) = 0. - R. J. Mathar, Nov 30 2012 [Valid for n >= 3; equivalently: a(n+1) = 2*n*a(n) for n > 1. - M. F. Hasler, Jan 15 2017]
G.f.: G(0) - 1, where G(k) = 1 + 1/(1 - 1/(1 + 1/((2*k + 2)*x*G(k+1)))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
Let s(n) = Sum_{k >= 1} 1/(2*k - 1)^n with n > 1, then s(n) = (-1)^n*PolyGamma(n-1, 1/2)/a(n). - Jean-François Alcover, Dec 18 2013
a(n) = round(-zeta(n)(1/2)) where zeta(n)(1/2) is the n-th derivative of the zeta function at 1/2. - Artur Jasinski, Feb 06 2021
E.g.f.: -x-log(1-2*x). - Alois P. Heinz, Mar 10 2022
Sum_{n>=1} 1/a(n) = (exp(1/2)+1)/2. - Amiram Eldar, Feb 02 2023

New name (using first formula) from Joerg Arndt, Mar 10 2022

A219570 Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 6, 6, 2, 6, 24, 36, 24, 6, 24, 120, 240, 240, 120, 24, 120, 720, 1800, 2400, 1800, 720, 120, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 5040, 40320, 141120, 282240, 352800, 282240, 141120, 40320, 5040, 40320, 362880, 1451520, 3386880, 5080320, 5080320, 3386880, 1451520, 362880, 40320

Offset: 0

Geoffrey Critzer, Nov 23 2012

Row sums are A066318.

			0;
1,      1;
1,      2,     1;
2,      6,     6,     2;
6,     24,    36,    24,     6;
24,   120,   240,   240,   120,    24;
120,  720,  1800,  2400,  1800,   720,  120;
720, 5040, 15120, 25200, 25200, 15120, 5040, 720;

Andrew Howroyd, Table of n, a(n) for n = 0..1274

nn=8;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Log[1/(1-(y+1)x)],{x,0,nn}],{x,y}],1]]//Grid

T(n, k) = if(n>0, (n-1)! * binomial(n, k)); \\ Andrew Howroyd, Oct 11 2017

E.g.f.: log(1/(1 - (y + 1)*x)).

T(n, k) = (n-1)! * binomial(n, k) for n > 0. - Andrew Howroyd, Oct 11 2017

A097632 a(n) = 2^n * Lucas(n) * (n-1)!.

Original entry on oeis.org

2, 12, 64, 672, 8448, 138240, 2672640, 60641280, 1568931840, 45705461760, 1478924697600, 52646746521600, 2044394156851200, 86005817907609600, 3896481847600742400, 189139342470414336000, 9793081532749971456000, 538748376721309827072000, 31381673358053118836736000

Offset: 1

Ralf Stephan, Aug 17 2004

nonn

Number of possible well-colored cycles on n nodes. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

A.H.M. Smeets, Table of n, a(n) for n = 1..100
C. Banderier, J.-M. Le Bars, and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Cf. A000079, A000142, A000204, A066318, A087131.

a[n_] := 2^n*LucasL[n,1]*(n-1)!; Array[a,19] (* or *)
nmax=19; CoefficientList[Series[-Log[1-2x-4x^2], {x,0,nmax}], x]Range[0,nmax]! (* Stefano Spezia, Jan 15 2024 *)

def A097632(n):
    L0, L1, F, i = 1, 2, 2, 1
    while i < n:
        L0, L1, F, i = L0+L1, L0, 2*i*F, i+1
    return L0*F # A.H.M. Smeets, Jan 15 2024

E.g.f.: -log(1-2*x-4*x^2).
a(n) = A000204(n) * A066318(n).
a(n) ~ sqrt(2*Pi/n)*(2*n*phi/e)^n. - Stefano Spezia, Jan 16 2024

Definition corrected by and a(18)-a(19) from Stefano Spezia, Jan 15 2024

A159749 The decomposition of a certain labeled universe (A052584), triangle read by rows.

Original entry on oeis.org

2, 2, 4, 2, 12, 16, 0, 24, 96, 96, -8, 0, 320, 960, 768, 0, -240, 0, 4800, 11520, 7680, 240, 0, -6720, 0, 80640, 161280, 92160, 0, 13440, 0, -188160, 0, 1505280, 2580480, 1290240, -24192, 0, 645120, 0, -5419008, 0, 30965760, 46448640, 20643840

Offset: 0

Peter Luschny, Apr 20 2009

T(n,k) is a weighted binomial sum of the Bernoulli numbers A027641/A027642 with A027641(1) = 1, which amounts to the definition B_{n} = B_{n}(1).

			2
2, 4
2, 12, 16
0, 24, 96, 96
-8, 0, 320, 960, 768
0, -240, 0, 4800, 11520, 7680
240, 0, -6720, 0, 80640, 161280, 92160

Cf. A027641, A027642, A052584.

T := (n,k) -> (n+1)!*binomial(n,k)*bernoulli(n-k,1)*2^(k+1)/(k+1);

T[n_, k_] := (n+1)! Binomial[n, k] BernoulliB[n-k, 1] 2^(k+1)/(k+1);
Table[T[n, k], {n, 0, 8}, {k, 0, n}] (* Jean-François Alcover, Jun 17 2019 *)

T(n,k) = (n+1)!*C(n,k)*B_{n-k}*2^(k+1)/(k+1).
T(n,n) = A066318(n+1) = n!*2^(n+1) (necklaces with n labeled beads of 2 colors; see also A032184).
Sum_{k=0..n} T(n,k) = A052584(n+1) = (n+1)!*(1+2^n).

A369881 Decimal expansion of 1/(2*sqrt(e)).

Original entry on oeis.org

3, 0, 3, 2, 6, 5, 3, 2, 9, 8, 5, 6, 3, 1, 6, 7, 1, 1, 8, 0, 1, 8, 9, 9, 7, 6, 7, 4, 9, 5, 5, 9, 0, 2, 2, 6, 7, 2, 0, 9, 5, 9, 0, 6, 7, 7, 4, 3, 5, 9, 3, 4, 7, 7, 8, 4, 1, 4, 4, 6, 0, 7, 9, 3, 6, 7, 5, 2, 8, 2, 5, 9, 7, 0, 6, 8, 7, 4, 2, 1, 1, 9, 9, 9, 3, 2, 3, 8, 0, 5, 7, 5, 3, 9, 9, 4, 7, 2, 8, 0, 1, 3, 2, 1, 1

Offset: 0

Amiram Eldar, Feb 04 2024

			0.30326532985631671180189976749559022672095906774359...

D. M. Batinetu-Giurgiu, Problem 764, The Pentagon, Vol. 75, No. 2 (2016), p. 22.
Toyesh Prakash Sharma, The Applications of the Stirling's approximation to find limits, Revista Electronica MateInfo.ro, December 2020, pp. 44-49. See Problem 13, p. 49.

Cf. A001113, A019774, A019778, A066318, A092605, A187832, A249455.

Mathematica
```
RealDigits[1/(2*Sqrt[E]), 10, 105][[1]]
```
PARI
```
exp(-1/2)/2
```

Equals A092605 / 2.
Equals exp(-(1 + A187832)).
Equals Sum_{n>=1} (-1)^(n+1)/A066318(n).
Equals lim_{n->oo} sqrt(n)*(((n+1)!)^(1/(2*(n+1))) - (n!)^(1/(2*n))) (Batinetu-Giurgiu, 2016).

cp's OEIS Frontend

A101925 a(n) = A005187(n) + 1.

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A032184 a(n) = 2^n*(n-1)! for n > 1, a(1) = 1.

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A219570 Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

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A097632 a(n) = 2^n * Lucas(n) * (n-1)!.

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Mathematica

Python

Formula

Extensions

A159749 The decomposition of a certain labeled universe (A052584), triangle read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

A369881 Decimal expansion of 1/(2*sqrt(e)).

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula