A001470 -id:A001470 - cp's OEIS frontend

A362309 Expansion of e.g.f. exp(x - x^3/3).

1, 1, 1, -1, -7, -19, 1, 211, 1009, 953, -14239, -105049, -209879, 1669669, 18057313, 56255291, -294375199, -4628130319, -19929569471, 70149241423, 1652969810521, 9226206209501, -20236475188159, -783908527648861, -5452368869656367

Offset: 0

Seiichi Manyama, Apr 15 2023

Seiichi Manyama, Table of n, a(n) for n = 0..632

Column k=2 of A362302.

Cf. A001470, A246607.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-x^3/3)))

a(n) = a(n-1) - 2 * binomial(n-1,2) * a(n-3) for n > 2.

a(n) = n! * Sum_{k=0..floor(n/3)} (-1/3)^k / (k! * (n-3*k)!).

A061138 Number of degree-n odd permutations of order exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 6, 30, 90, 210, 1680, 12096, 114660, 833580, 5928120, 38112360, 259194936, 1739195640, 17043237120, 167089937280, 1837707369840, 18342985021776, 181206905922720, 1673742164139360, 16992525855006240

Offset: 0

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.

E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4).

A061139 Number of degree-n odd permutations of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 20, 240, 1260, 5600, 45360, 383040, 2451680, 17128320, 157769040, 1902380480, 18882623760, 163633317120, 2095059774080, 30792478993920, 346562329685760, 3905491275514880, 58609449249207360, 866031730098205440

Offset: 0

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.

E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).

A381975 Number of ways for n competitors to rank in a competition in which each match has 4 possible outcomes in which each competitor gains 0, 1, 2 or 3 points.

Original entry on oeis.org

1, 1, 2, 9, 58, 459, 4370, 48999, 632884, 9254473, 151155362, 2727862751

Offset: 0

SiYang Hu, May 06 2025

Also, the number of maps f:{1, 2, ..., n} -> {1, 2, ..., n} such that f(f(f(x))) >= x for all 1 <= x <= n.

When this definition is changed to f(f(x)), then the result would be the Fubini numbers (A000670).

Cf. A000142, A000312, A000670, A001470.

validMappings :: Int -> Int
validMappings n = validMappingsDFS n [0] 1 where
  checkCondition f = all (\x -> f !! (f !! (f !! (x-1)) - 1) - 1 >= x) [1..n]
  validMappingsDFS n f x
    | x > n = if checkCondition f then 1 else 0
    | otherwise = sum [validMappingsDFS n (take (x-1) f ++ [i] ++ drop x f) (x+1) | i <- [1..n]]

CheckCondition[f_, n_] := AllTrue[Range[n], (f[[f[[f[[#]]]]]] >= # &)]
validMappingsDFS[n_, f_, x_] := Module[{i, f2, count},
  If[x > n,
    If[CheckCondition[f, n], 1, 0],
    count = 0;
    For[i = 1, i <= n, i++,
      f2 = f; f2[[x]] = i;  (* Assign mapping for f[x] *)
      count += validMappingsDFS[n, f2, x + 1];  (* Explore further *)
    ];
    count
  ]
]
A381975[n_] := validMappingsDFS[n, ConstantArray[0, n], 1]
Table[A381975[n], {n, 0, 6}]

def validMappings(n):
    def checkCondition(f, n):
        return all(f[f[f[x]]] >= x for x in range(1, n+1))
    def validMappingsDFS(n, f, x):
        if x > n:
            return 1 if checkCondition(f, n) else 0
        return sum(validMappingsDFS(n, f[:x] + [i] + f[x+1:], x+1) for i in range(1, n+1))
    return validMappingsDFS(n, [0] * (n+1), 1)

validMappings <- function(n) {
  checkCondition <- function(f, n) all(sapply(1:n, function(x) f[f[f[x]]] >= x))
  validMappingsDFS <- function(n, f, x) {
    if (x > n) return(ifelse(checkCondition(f, n), 1, 0))
    sum(sapply(1:n, function(i) { f[x] <- i; validMappingsDFS(n, f, x + 1) }))
  }
  validMappingsDFS(n, rep(0, n), 1)
}

a(11) from Sean A. Irvine, May 13 2025

A061122 Number of degree-n permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 45360, 453600, 3326400, 39916800, 363242880, 3874590720, 34767532800, 567177811200, 6897521030400, 98241008785920, 1138935652807680, 18952720774041600, 258251731634534400

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

E.g.f.: -exp(x+1/2*x^2+1/4*x^4)+exp(x+1/2*x^2+1/4*x^4+1/8*x^8).

A061123 Number of degree-n permutations of order exactly 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 403200, 2217600, 26611200, 259459200, 1695133440, 16345929600, 161902540800, 1208560953600, 50830132953600, 866513503215360, 8470676211379200, 166891791625977600, 2699606616475507200

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

E.g.f.: -exp(x+1/3*x^3)+exp(x+1/3*x^3+1/9*x^9).

A061124 Number of degree-n permutations of order exactly 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 504, 4032, 27216, 514080, 4823280, 57081024, 500972472, 4412103696, 60619398840, 686638592640, 9335025764064, 104304736815552, 1180585704051936, 29016515871665280, 478096386437121480

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

E.g.f.: exp(x) - exp(x+1/2*x^2) - exp(x+1/5*x^5) + exp(x+1/2*x^2+1/5*x^5+1/10*x^10).
From Benedict W. J. Irwin, May 27 2016: (Start)
Let y1(0)=1, y1(1)=1,
Let -y1(n)-y1(n+1)+(n+2)*y1(n+2)=0,
Let y2(0)=1, y2(1)=1, y2(2)=1/2, y2(3)=1/6, y2(4)=1/24,
Let -y2(n)-y2(n+4)+(n+5)*y2(n+5)=0,
Let y3(0)=1, y3(1)=1, y3(2)=1, y3(3)=2/3, y3(4)=5/12, y3(5)=5/12, y3(6)=11/36, y3(7)=31/126, y3(8)=307/2016, y3(9)=1643/18144,
Let -y3(n)-y3(n+5)-y3(n+8)-y3(n+9)+(n+10)*y3(n+10)=0,
a(n) = 1+n!*(y3(n)-y2(n)-y1(n)).
(End)

A061125 Number of degree-n permutations of order exactly 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 420, 3360, 30240, 403200, 4019400, 80166240, 965284320, 12173441280, 162850287600, 2428557331200, 32123543612160, 534678700308480, 8126981741380320, 128338880777251200, 2080312367956502400, 36351373041072122880, 606331931399062693440

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

nn=21;Range[0,nn]!CoefficientList[Series[(Exp[x^12/12]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^6/6]-1)(Exp[x^4/4]-1)Exp[x+x^2/2+x^3/3]+(Exp[x^4/4]-1)(Exp[x^3/3]-1)Exp[x^2/2+x],{x,0,nn}],x]//Rest  (* Geoffrey Critzer, Feb 04 2013 *)

E.g.f.: exp(x + 1/2*x^2) - exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12).

A061126 Number of degree-n permutations of order exactly 16.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1307674368000, 22230464256000, 400148356608000, 5068545850368000, 101370917007360000, 1490152480008192000, 24977793950613504000, 343667682838351872000

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

E.g.f.: - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8 + 1/16*x^16).

A061127 Number of degree-n permutations of order exactly 24.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1663200, 19958400, 259459200, 4843238400, 72648576000, 988020633600, 14600749363200, 224704121241600, 3614691131251200, 84808750650624000, 1509309706083379200, 29359195162807910400

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x^24/24]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6+x^8/8+x^12/12]+(Exp[x^12/12]-1)(Exp[x^8/8]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^8/8]-1)(Exp[x^6/6]-1)Exp[x+x^2/2+x^3/3+x^4/4]+(Exp[x^8/8]-1)(Exp[x^3/3]-1)Exp[x+x^2/2+x^4/4],{x,0,nn}],x]//Rest (* Geoffrey Critzer, Feb 04 2013 *)

E.g.f.: exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/8*x^8 + 1/12*x^12 + 1/24*x^24).

cp's OEIS Frontend

A362309 Expansion of e.g.f. exp(x - x^3/3).

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A061138 Number of degree-n odd permutations of order exactly 4.

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A061139 Number of degree-n odd permutations of order exactly 6.

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A381975 Number of ways for n competitors to rank in a competition in which each match has 4 possible outcomes in which each competitor gains 0, 1, 2 or 3 points.

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A061122 Number of degree-n permutations of order exactly 8.

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A061123 Number of degree-n permutations of order exactly 9.

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A061124 Number of degree-n permutations of order exactly 10.

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A061125 Number of degree-n permutations of order exactly 12.

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A061126 Number of degree-n permutations of order exactly 16.

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A061127 Number of degree-n permutations of order exactly 24.

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