A000392 -id:A000392 - cp's OEIS frontend

A056279 Number of primitive (aperiodic) word structures of length n which contain exactly three different symbols.

0, 0, 1, 6, 25, 89, 301, 960, 3024, 9305, 28501, 86430, 261625, 788669, 2375075, 7140720, 21457825, 64435896, 193448101, 580597110, 1742343323, 5228050949, 15686335501, 47063113320, 141197991000, 423610488665, 1270865802276, 3812663735790, 11438127792025, 34314649427035

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column 3 of A137651.

Cf. A056268.

a(n) = Sum_{d|n} mu(d)*A000392(n/d) where n>0.

G.f.: Sum_{k>=1} mu(k) * x^(3*k) / Product_{j=1..3} (1 - j*x^k). - Ilya Gutkovskiy, Apr 15 2021

Terms a(26) and beyond from Andrew Howroyd, Apr 15 2021

A133800 Triangle read by rows in which row n gives number of ways to partition n labeled elements into k pie slices allowing the pie to be turned over (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 3, 1, 15, 25, 30, 12, 1, 31, 90, 195, 180, 60, 1, 63, 301, 1050, 1680, 1260, 360, 1, 127, 966, 5103, 12600, 15960, 10080, 2520, 1, 255, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 1, 511, 9330, 102315, 510300, 1369620

Offset: 1

Barry Cipra and N. J. A. Sloane, Jan 17 2008

			Triangle begins:
1,
1,  1,
1,  3,   1,
1,  7,   6,    3,
1, 15,  25,   30,   12,
1, 31,  90,  195,  180,   60,
1, 63, 301, 1050, 1680, 1260, 360.
...
For row n = 4 we have the following "pies":
. 1
./ \
2 . 3 . 12 .. 12 . 123
.\ / .. / \ .(..)..(..)
. 4 .. 3--4 . 34 .. 4 .. (1234)
k=4 .. k=3 ..k=2 . k=2 . k=1
(3)....(6)...(3)..(4)... (1)

C. G. Bower, Transforms (2)

Row sums give A032262. Diagonals give A000225, A000392, A032263, A133799, A001710.

A001710 := proc(n) if n < 2 then 1; else n!/2 ; fi ; end: A008277 := proc(n,k) combinat[stirling2](n,k) ; end: A133800 := proc(n,k) A008277(n,k)*A001710(k-1) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d, ",A133800(n,k)) ; od: od: # R. J. Mathar, Jan 18 2008

A001710[n_] := If[n<2, 1, n!/2]; A008277[n_, k_] := StirlingS2[n, k]; A133800[n_, k_] := A008277[n, k]*A001710[k-1]; Table[A133800[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014, after R. J. Mathar *)
(* A (n >= 0, k >= 0)-based version: *)
A133800[n_, k_] := k! StirlingS2[n+1, k+1] / If[k>1, 2, 1];
Table[A133800[n,k], {n,0,9}, {k,0,n}] // Flatten (* Peter Luschny, Oct 19 2017 *)

Take triangle of Stirling numbers of second kind (A008277) and multiply k-th column by A001710(k) (order of alternating group A_k).

More terms from R. J. Mathar, Jan 18 2008

A249163 Triangle read by rows: the positive terms of A163626.

Original entry on oeis.org

1, 1, 1, 2, 1, 12, 1, 50, 24, 1, 180, 360, 1, 602, 3360, 720, 1, 1932, 25200, 20160, 1, 6050, 166824, 332640, 40320, 1, 18660, 1020600, 4233600, 1814400, 1, 57002, 5921520, 46070640, 46569600, 3628800, 1, 173052, 33105600, 451725120, 898128000, 239500800

Offset: 0

Paul Curtz, Dec 15 2014

We have two possibilities: with or without 0's.
Without 0's:
1,
1,
1,   2,
1,  12,
1,  50,  24,
1, 180, 360,
etc.
Sum of every row: A000670(n).
First two terms of successive columns: 1, 1, 2, 12, 24, 360, ... = A211374.
With 0's:
1,   0,    0,   0,
1,   0,    0,   0,
1,   2,    0,   0,
1,  12,    0,   0,
1,  50,   24,   0,
1, 180,  360,   0,
1, 602, 3360, 720,
etc.
The columns are essentially A000012, A028243, A028246, A228909, A228911, A228913, from Stirling numbers of the second kind S(n,3), S(n,5), S(n,7), S(n,9), S(n,11), ... .

Cf. A163626, A000670, A211374; also A000012, A000392, A000481, A000771, A049447, A028243, A028246, A091137, A228909, A163626, A228911, A228913 and Worpitzky numbers for the second Bernoulli numbers A164555(n)/A027642(n).

Derivative[0][y][x] = y[x]; Derivative[1][y][x] = y[x]*(1 - y[x]); Derivative[n_][y][x] := Derivative[n][y][x] = D[Derivative[n - 1][y][x], x]; row[n_] := CoefficientList[Derivative[n][y][x], y[x]] // Rest; Table[ Select[row[n], Positive] , {n, 0, 12}] // Flatten
(* or, simply: *) Table[(-1)^k*k!*StirlingS2[n+1, k+1], {n, 0, 12}, {k, 0, n}] // Flatten // Select[#, Positive]& (* Jean-François Alcover, Dec 16 2014 *)

A249999 Expansion of 1/((1-x)^2(1-2x)(1-3x)).

Original entry on oeis.org

1, 7, 32, 122, 423, 1389, 4414, 13744, 42245, 128771, 390396, 1179366, 3554467, 10696153, 32153978, 96592988, 290041089, 870647535, 2612991160, 7841070610, 23527406111, 70590606917, 211788597942, 635399348232, 1906265153533, 5718929678299, 17157057470324, 51471709281854

Offset: 0

Alex Ratushnyak, Dec 28 2014

G. C. Greubel, Table of n, a(n) for n = 0..1000
Anthony G. Shannon, Hakan Akkuş, Yeşim Aküzüm, Ömür Deveci, and Engin Özkan, A partial recurrence Fibonacci link, Notes Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 530-537. See Table 2, p. 534.
Index entries for linear recurrences with constant coefficients, signature (7,-17,17,-6).

Cf. A000392 (first differences), A094705, A243869, A249997.

[(2*n +9 -2^(n+5) +3^(n+3))/4: n in [0..50]]; // G. C. Greubel, Jul 21 2022

LinearRecurrence[{7,-17,17,-6}, {1,7,32,122}, 50] (* G. C. Greubel, Jul 21 2022 *)
CoefficientList[Series[1/((1-x)^2(1-2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, Feb 11 2025 *)

[(2*n+9 -2^(n+5) +3^(n+3))/4 for n in (0..50)] # G. C. Greubel, Jul 21 2022

G.f.: 1/((1-x)^2 * (1-2*x) * (1-3*x)).
a(n) = 9/4 - 2^(n+3) + n/2 + 3^(n+3)/4. - R. J. Mathar, Jan 09 2015
E.g.f.: (1/4)*((9 + 2*x) - 32*exp(x) + 27*exp(2*x))*exp(x). - G. C. Greubel, Jul 21 2022

A250118 Triangle read by rows: T(n,m) (n >= 1, 1 <= m <= n) = number of set partitions of [n], avoiding 12343, with m blocks.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 9, 1, 1, 31, 90, 52, 12, 1, 1, 63, 301, 246, 88, 15, 1, 1, 127, 966, 1039, 510, 133, 18, 1, 1, 255, 3025, 4083, 2569, 909, 187, 21, 1, 1, 511, 9330, 15274, 11790, 5296, 1470, 250, 24, 1, 1, 1023, 28501, 55152, 50644, 27678, 9706, 2220, 322, 27, 1

Offset: 1

N. J. A. Sloane, Nov 25 2014

			Triangle begins:
  1;
  1,    1;
  1,    3,     1;
  1,    7,     6,     1;
  1,   15,    25,     9,     1;
  1,   31,    90,    52,    12,     1;
  1,   63,   301,   246,    88,    15,    1;
  1,  127,   966,  1039,   510,   133,   18,    1;
  1,  255,  3025,  4083,  2569,   909,  187,   21,   1;
  1,  511,  9330, 15274, 11790,  5296, 1470,  250,  24,  1;
  1, 1023, 28501, 55152, 50644, 27678, 9706, 2220, 322, 27, 1;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..5050 (The first 100 rows.)
Harry Crane, Left-right arrangements, set partitions, and pattern avoidance, Australasian Journal of Combinatorics, 61(1) (2015), 57-72.

Cf. A112857, A250119. For diagonals see A000392, A163941.

a(46)-a(66) from Lars Blomberg, Aug 17 2017

A260217 Number of base-3 n-digit pandigital numbers.

Original entry on oeis.org

0, 0, 4, 24, 100, 360, 1204, 3864, 12100, 37320, 114004, 346104, 1046500, 3155880, 9500404, 28566744, 85831300, 257756040, 773792404, 2322425784, 6969374500, 20912317800, 62745342004, 188252803224, 564791964100, 1694443001160, 5083463221204, 15250658099064

Offset: 1

Jon E. Schoenfield, Jul 19 2015

From Manfred Boergens, Aug 02 2023: (Start)
a(n+1) is the number of pairs (A,B) where A and B are nonempty subsets of {1,2,...,n} and one of these subsets is a proper subset of the other.
If "proper" is omitted, see A091344.
If empty subsets are included, see A027649 (all subsets) and A056182 (proper subsets). (End)

			a(3)=4 because, in base 3, there are four 3-digit pandigital numbers (11=102_3, 15=120_3, 19=201_3, and 21=210_3).
a(4)=24 because, in base 3, there are 24 4-digit pandigital numbers (1002_3, 1012_3, 1020_3, 1021_3, 1022_3, 1102_3, 1120_3, 1200_3, 1201_3, 1202_3, 1210_3, 1220_3, 2001_3, 2010_3, 2011_3, 2012_3, 2021_3, 2100_3, 2101_3, 2102_3, 2110_3, 2120_3, 2201_3, and 2210_3).

Svenja Huntemann, Values, Temperatures, and Enumeration of Placement Games, Slides, Alberta-Montana Combinatorics and Algorithms Day, Banff, Canada, 23-25 June 2023. See p. 105/109.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A000392, A027649, A028243, A056182, A091344, A171102.

[2*3^(n-1) - 2^(n+1) + 2: n in [1..30]]; // Vincenzo Librandi, Jul 20 2015

Table[2 3^(n - 1) - 2^(n + 1) + 2, {n, 30}] (* Vincenzo Librandi, Jul 20 2015 *)

a(n) = 2*A028243(n) = 2*3^(n-1) - 2^(n+1) + 2.
a(n) = 4*A000392(n).
G.f.: 4*x^3/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: 2/3*((exp(x)-1)^3).

A327610 Number of length n reversible string structures that are not palindromic using exactly three different colors.

Original entry on oeis.org

0, 0, 1, 4, 14, 49, 154, 496, 1520, 4705, 14266, 43384, 130844, 394849, 1187614, 3571936, 10729040, 32222785, 96724306, 290313064, 871172324, 2614069249, 7843168774, 23531688976, 70598997560, 211805640865, 635432906746, 1906333059544, 5719063904204

Offset: 1

Andrew Howroyd, Sep 18 2019

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (6,-6,-24,49,6,-66,36).

Column k=3 of A309748.

Cf. A000392, A056327, A284949, A327611.

concat([0,0], Vec((1 - 2*x - 4*x^2 + 13*x^3 - 9*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 2*x^2)*(1 - 3*x^2)) + O(x^30)))

a(n) = A056327 - A000392(ceiling(n/2)).
a(n) = 6*a(n-1) - 6*a(n-2) - 24*a(n-3) + 49*a(n-4) + 6*a(n-5) - 66*a(n-6) + 36*a(n-7) for n > 7.
G.f.: x^3*(1 - 2*x - 4*x^2 + 13*x^3 - 9*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 2*x^2)*(1 - 3*x^2)).

A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

Original entry on oeis.org

0, 0, 648, 3888, 16200, 58320, 195048, 625968, 1960200, 6045840, 18468648, 56068848, 169533000, 511252560, 1539065448, 4627812528, 13904670600, 41756478480, 125354369448, 376232977008, 1129038669000, 3387795483600, 10164745404648, 30496954122288, 91496298184200

Offset: 1

Stefano Spezia, Aug 22 2020

a(n) is the number of n-digit numbers in A031962.

			a(1) = a(2) = 0 since the positive integers must have at least three digits;
a(3) = #{xyz in N | x,y,z are three different digits with x != 0} = 9*9*8 = 648;
a(4) = 3888 since #[9999] - #[999] - #(1111*[9]) - A335843(4) - #{xywz in N | x,y,w,z are four different digits with x != 0} = 9999 - 999 - 9 - 567 - 9*9*8*7 = 3888;
...

Index entries for linear recurrences with constant coefficients, signature (6,-11,6).
Index entries for sequences related to digits.

Cf. A000225, A000244, A000392, A031962, A048993, A335843, A337127.

Cf. A055642, A180599, A235155, A235691, A235718.

LinearRecurrence[{6,-11,6},{0,0,648},26]

concat([0,0],Vec(648*x^3/(1-6*x+11*x^2-6*x^3)+O(x^26)))

O.g.f.: 648*x^3/(1 - 6*x + 11*x^2 - 6*x^3).
E.g.f.: 108*(exp(x) - 1)^3.
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n > 3.
a(n) = 648*S2(n, 3) where S2(n, 3) = A000392(n).
a(n) = 324*(3^(n-1) - 2^n + 1).
a(n) ~ 108 * 3^n.
a(n) = 324*(A000244(n-1) - A000225(n)).
a(n) = A337127(n, 3).

A346390 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^3 / 3! ).

Original entry on oeis.org

1, 6, 25, 100, 511, 3626, 30045, 262800, 2470171, 25889446, 302003065, 3821936300, 51672723831, 745789322466, 11505096936085, 189023074558600, 3288243760145491, 60319276499454686, 1164282909466221105, 23603464830964817700, 501435697062735519151

Offset: 3

Ilya Gutkovskiy, Aug 08 2021

nonn

Cf. A000392, A000629, A003704, A327504, A346894, A346954, A346955.

nmax = 23; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^3/3!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 3] &
a[n_] := a[n] = StirlingS2[n, 3] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 3, 23}]

my(x='x+O('x^25)); Vec(serlaplace(-log(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 09 2021

a(n) = Stirling2(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,3) * k * a(k).
a(n) ~ (n-1)! / (log(6^(1/3)+1))^n. - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/3)} (3*k)! * Stirling2(n,3*k)/(k * 6^k). - Seiichi Manyama, Jan 23 2025

A052761 a(n) = 3!nS(n-1,3), where S denotes the Stirling numbers of second kind.

Original entry on oeis.org

0, 0, 0, 0, 24, 180, 900, 3780, 14448, 52164, 181500, 615780, 2052072, 6749028, 21976500, 71007300, 228009696, 728451972, 2317445100, 7346047140, 23213772120, 73156412196, 229989358500, 721474964100, 2258832312144, 7059480120900, 22026886599900

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Matthew House, Table of n, a(n) for n = 0..2079
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 718
Index entries for linear recurrences with constant coefficients, signature (12,-58,144,-193,132,-36).

Cf. A000392, A001117, A052749.

spec := [S,{B=Set(Z,1 <= card),S=Prod(B,B,B,Z)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0},Table[3!*n*StirlingS2[n-1,3],{n,30}]] (* Harvey P. Dale, Feb 07 2015 *)

a(n)={if(n>=1, 3!*n*stirling(n-1, 3, 2), 0)} \\ Andrew Howroyd, Aug 08 2020

E.g.f.: exp(x)^3*x - 3*exp(x)^2*x + 3*x*exp(x) - x.
Recurrence: {a(1)=0, a(2)=0, a(3)=0, a(4)=24, (-36*n^2 - 66*n - 6*n^3 - 36)*a(n) + (11*n^3 + 55*n^2 + 66*n)*a(n+1) + (-6*n^3 - 24*n^2 - 18*n)*a(n+2) + (n^3 + 3*n^2 + 2*n)*a(n+3)}
For n>=2, a(n) = n*(3^(n-1) - 3*2^(n-1) + 3). - Vaclav Kotesovec, Nov 27 2012
O.g.f.: 12*x^4*(2 - 9*x + 11*x^2 - 3*x^3)/((1 - 3*x)^2*(1 - 2*x)^2*(1 - x)^2). - Matthew House, Feb 16 2017 [Corrected by Georg Fischer, May 19 2019]
From Andrew Howroyd, Aug 08 2020: (Start)
a(n) = n*A001117(n-1) for n > 1.
E.g.f.: x*(exp(x) - 1)^3. (End)

Better description from Victor Adamchik (adamchik(AT)cs.cmu.edu), Jul 19 2001

More terms from Harvey P. Dale, Feb 07 2015

cp's OEIS Frontend

A056279 Number of primitive (aperiodic) word structures of length n which contain exactly three different symbols.

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A133800 Triangle read by rows in which row n gives number of ways to partition n labeled elements into k pie slices allowing the pie to be turned over (n >= 1, 1 <= k <= n).

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Maple

Mathematica

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A249163 Triangle read by rows: the positive terms of A163626.

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Mathematica

A249999 Expansion of 1/((1-x)^2*(1-2*x)*(1-3*x)).

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Magma

Mathematica

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A250118 Triangle read by rows: T(n,m) (n >= 1, 1 <= m <= n) = number of set partitions of [n], avoiding 12343, with m blocks.

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A260217 Number of base-3 n-digit pandigital numbers.

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Magma

Mathematica

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A327610 Number of length n reversible string structures that are not palindromic using exactly three different colors.

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PARI

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A337313 a(n) is the number of n-digit positive integers with exactly three distinct base 10 digits.

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A249999 Expansion of 1/((1-x)^2(1-2x)(1-3x)).

A052761 a(n) = 3!nS(n-1,3), where S denotes the Stirling numbers of second kind.