A000255 -id:A000255 - cp's OEIS frontend

A350313 The Redstone permutation: a(1) = 2, a(2) = 1, otherwise the smallest number not occurring earlier which is strongly prime to n.

2, 1, 4, 5, 3, 7, 8, 9, 10, 11, 6, 13, 14, 15, 16, 17, 12, 19, 20, 21, 22, 23, 18, 25, 26, 27, 28, 29, 24, 31, 32, 33, 34, 35, 36, 37, 30, 39, 40, 41, 38, 43, 44, 45, 46, 47, 42, 49, 50, 51, 52, 53, 48, 55, 56, 57, 58, 59, 54, 61, 62, 63, 64, 65, 66, 67, 60, 69, 70

Offset: 1

Peter Luschny, Dec 24 2021

nonn

We say 'k is strongly prime to n' if and only if k is prime to n and k does not divide n - 1 (see A181830).
The sequence is a fixed-point-free permutation of the positive integers beginning with 2. According to Don Knuth, the number of fixed-point-free permutations beginning with 2 of [n] = {1, 2, ..., n} were already computed by Euler, see A000255.
We say n is a 'catch-up point' of a permutation p of the positive integers if and only if p restricted to [n] is a permutation of [n]. The catch-up points of this sequence start 2, 5, 11, 17, ... and are in A350314. This structure allows the sequence to be seen as an irregular triangle, as shown in the example section. The lengths of the resulting rows are a periodic sequence (see A350315).

			Catch-up points and initial segments:
[ 2]  2,  1,
[ 5]  4,  5,  3,
[11]  7,  8,  9, 10, 11,  6,
[17] 13, 14, 15, 16, 17, 12,
[23] 19, 20, 21, 22, 23, 18,
[29] 25, 26, 27, 28, 29, 24,
[37] 31, 32, 33, 34, 35, 36, 37, 30,
[41] 39, 40, 41, 38,
[47] 43, 44, 45, 46, 47, 42,
[53] 49, 50, 51, 52, 53, 48,
...

Cf. A350314, A350315, A181830, A000255.

s = {2, 1}, c[] = 0; Array[Set[c[s[[#]]], #] &, Length[s]]; j = Last[s]; u = 3; s~Join~Reap[Monitor[Do[If[j == u, While[c[u] > 0, u++]]; k = u; While[Nand[c[k] == 0, CoprimeQ[i, k], ! Divisible[i - 1, k]], k++]; Sow[k]; Set[c[k], i]; j = k, {i, Length[s] + 1, 69}], i]][[-1, -1]] (* _Michael De Vlieger, Dec 24 2021 *)

def generatePermutation(N, condition):
    a = {1:2, 2:1}; n = Integer(2)
    notYetOccured = [Integer(i) for i in range(3, N + 1)]
    while notYetOccured != []:
        n += 1
        found = False
        for r in notYetOccured:
            if condition(r, n):
                a[n] = r
                notYetOccured.remove(r)
                found = True
                break
        if not found: break
    return [a[i] for i in range(1, n)]
def isPrimeTo(m, n): return gcd(m, n) == 1
def isStrongPrimeTo(m, n): return isPrimeTo(m, n) and not m.divides(n - 1)
print(generatePermutation(70, isStrongPrimeTo))

A370383 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4].

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 717, 5026, 40242, 362376, 3625081, 39885851, 478714416, 6224078292, 87145277160, 1307271652917, 20917481850667, 355612235468396, 6401234296266540, 121626707638142280, 2432586885636105251, 51085230669413519349, 1123891538655073251190

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.

Cf. A000255, A002628, A132647, A184182, A370384.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*((x-x^5)/(1-x^5))^k))

G.f.: Sum_{k>=0} k! * ( (x-x^5)/(1-x^5) )^k.

a(n) = Sum_{k=0..4} A184182(n,k). - Alois P. Heinz, Feb 17 2024

A370384 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4,k+5].

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5037, 40306, 362802, 3628296, 39913080, 478970641, 6226733531, 87175347936, 1307641346772, 20922387099240, 355682119243320, 6402298503373917, 121643960874649867, 2432883613692550316, 51090627024035616300, 1123995015882951892680

Offset: 0

Seiichi Manyama, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.

Cf. A000255, A002628, A132647, A184182, A370383.

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, k!*((x-x^6)/(1-x^6))^k))

G.f.: Sum_{k>=0} k! * ( (x-x^6)/(1-x^6) )^k.

a(n) = Sum_{k=0..5} A184182(n,k). - Alois P. Heinz, Feb 17 2024

A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

Original entry on oeis.org

1, 2, 8, 40, 248, 1808, 15136, 142784, 1496960, 17254144, 216740864, 2945973248, 43065951232, 673626675200, 11224114860032, 198447384666112, 3710328985124864, 73136238041563136, 1515739708283944960, 32947698735175172096, 749499782353468522496, 17806903161183314378752

Offset: 0

Seiichi Manyama, Apr 24 2025

Cf. A000261, A383344, A383378.
Cf. A000023, A052124, A087981, A383381.
Cf. A000255.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^4))

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A000255.
a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+3,3)/(n-k)!.
a(0) = 1, a(1) = 2; a(n) = (n+1)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(Pi) * n^(n + 7/2) / (3*sqrt(2)*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A180185 Triangle read by rows: T(n,k) is the number of permutations of [n] having no 3-sequences and having k successions (0 <= k <= floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 11, 9, 1, 53, 44, 9, 309, 265, 66, 3, 2119, 1854, 530, 44, 16687, 14833, 4635, 530, 11, 148329, 133496, 44499, 6180, 265, 1468457, 1334961, 467236, 74165, 4635, 53, 16019531, 14684570, 5339844, 934472, 74165, 1854, 190899411

Offset: 0

Emeric Deutsch, Sep 06 2010

Row n has 1+floor(n/2) entries.

Sum of entries in row n is A002628(n).

			T(6,3)=3 because we have 125634, 341256, and 563412.
Triangle starts:
     1;
     1;
     1,    1;
     3,    2;
    11,    9,    1;
    53,   44,    9;
   309,  265,   66,    3;
  2119, 1854,  530,   44;

Cf. A000166, A002628, A002629, A000255.

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n, k) if n = 0 and k = 0 then 1 elif k <= (1/2)*n then binomial(n-k, k)*d[n+1-k]/(n-k) else 0 end if end proc: for n from 0 to 12 do seq(a(n, k), k = 0 .. (1/2)*n) end do; # yields sequence in triangular form

d[0] = 1; d[n_] := d[n] = n d[n - 1] + (-1)^n;
T[n_, k_] := If[n == 0 && k == 0, 1, If[k <= n/2, Binomial[n - k, k] d[n + 1 - k]/(n - k), 0]];
Table[T[n, k], {n, 0, 20}, {k, 0, Quotient[n, 2]}] // Flatten (* Jean-François Alcover, May 23 2020 *)

d(n) = if(n<2, !n , round(n!/exp(1)));
for(n=0, 20, for(k=0, (n\2), print1(binomial(n - k, k)*(d(n - k) + d(n - k - 1)),", ");); print();) \\ Indranil Ghosh, Apr 12 2017

T(n,k) = binomial(n-k,k)*(d(n-k) + d(n-k-1)), where d(j) = A000166(j) are the derangement numbers.
T(n,0) = d(n) + d(n-1) = A000255(n-1).
T(n,1) = d(n).
Sum_{k>=0} k*T(n,k) = A002629(n+1).

A195326 Numerators of fractions leading to e - 1/e (A174548).

Original entry on oeis.org

0, 2, 2, 7, 7, 47, 47, 5923, 5923, 426457, 426457, 15636757, 15636757, 7318002277, 7318002277, 1536780478171, 1536780478171, 603180793741, 603180793741, 142957467201379447, 142957467201379447

Offset: 0

Paul Curtz, Oct 12 2011

The sequence of approximations of exp(1) obtained by truncating the Taylor series of exp(x) after n terms is A061354(n)/A061355(n) = 1, 2, 5/2, 8/3, 65/24, ...
A Taylor series of exp(-1) is 1, 0, 1/2, 1/3, 3/8, ... and (apart from the first 2 terms) given by A000255(n)/A001048(n). Subtracting both sequences term by term we obtain a series for exp(1) - exp(-1) = 0, 2, 2, 7/3, 7/3, 47/20, 47/20, 5923/2520, 5923/2520, 426457/181440, 426457/181440, ... which defines the numerators here.
Each second of the denominators (that is 3, 2520, 19958400, ...) is found in A085990 (where each third term, that is 60, 19958400, ...) is to be omitted.
This numerator sequence here is basically obtained by doubling entries of A051397, A009628, A087208, or A186763, caused by the standard associations between cosh(x), sinh(x) and exp(x).

			a(0) =  1  -  1;
a(1) =  2  -  0;
a(2) = 5/2 - 1/2.

Cf. A001113, A068985, A197222, A197223.

taylExp1 := proc(n)
        add(1/j!,j=0..n) ;
end proc:
A000255 := proc(n)
        if n <=1 then
                1;
        else
                n*procname(n-1)+(n-1)*procname(n-2) ;
        end if;
end proc:
A001048 := proc(n)
        n!+(n-1)! ;
end proc:
A195326 := proc(n)
        if n = 0 then
                0;
        elif n =1 then
                2;
        else
                taylExp1(n) -A000255(n-2)/A001048(n-1);
        end if;
          numer(%);
end proc:
seq(A195326(n),n=0..20) ; # R. J. Mathar, Oct 14 2011

Material meant to be placed in other sequences removed by R. J. Mathar, Oct 14 2011

A218538 Triangle read by rows: T(n,k) is the number of permutations of{1,2,...,n} avoiding [x,x+1] having genus k (see first comment for definition of genus).

Original entry on oeis.org

1, 1, 0, 3, 0, 0, 7, 4, 0, 0, 19, 29, 5, 0, 0, 53, 180, 76, 0, 0, 0, 153, 1004, 901, 61, 0, 0, 0, 453, 5035, 8884, 2315, 0, 0, 0, 0, 1367, 23653, 74177, 46285, 2847, 0, 0, 0, 0, 4191, 106414, 546626, 667640, 143586, 0, 0, 0, 0, 0, 13015, 463740, 3658723, 7777935, 3896494, 209624, 0

Offset: 1

Joerg Arndt, Nov 01 2012

The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q.
Row sums are A000255 (permutations with no substring [x,x+1]).
First column is A078481.

			Triangle starts:
[ 1]  1,
[ 2]  1, 0,
[ 3]  3, 0, 0,
[ 4]  7, 4, 0, 0,
[ 5]  19, 29, 5, 0, 0,
[ 6]  53, 180, 76, 0, 0, 0,
[ 7]  153, 1004, 901, 61, 0, 0, 0,
[ 8]  453, 5035, 8884, 2315, 0, 0, 0, 0,
[ 9]  1367, 23653, 74177, 46285, 2847, 0, 0, 0, 0,
[10]  4191, 106414, 546626, 667640, 143586, 0, 0, 0, 0, 0,
[11]  13015, 463740, 3658723, 7777935, 3896494, 209624, 0, 0, 0, 0, 0,
[12]  40857, 1972339, 22712736, 77535694, 74678363, 13959422, 0, 0, ...,
[13]  129441, 8228981, 132804891, 685673340, 1131199122, 485204757, 23767241, 0, ...,
...

Cf. A177267 (genus of all permutations).

Cf. A178514 (genus of derangements), A178515 (genus of involutions), A178516 (genus of up-down permutations), A178517 (genus of non-derangement permutations), A178518 (permutations of [n] having genus 0 and p(1)=k), A185209 (genus of connected permutations).

A247490 Square array read by antidiagonals: A(k, n) = (-1)^(n+1)* hypergeom([k, -n+1], [], 1) for n>0 and A(k,0) = 0 (n>=0, k>=1).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 2, 3, 2, 0, 1, 3, 7, 11, 9, 0, 1, 4, 13, 32, 53, 44, 0, 1, 5, 21, 71, 181, 309, 265, 0, 1, 6, 31, 134, 465, 1214, 2119, 1854, 0, 1, 7, 43, 227, 1001, 3539, 9403, 16687, 14833, 0, 1, 8, 57, 356, 1909, 8544, 30637, 82508, 148329, 133496

Offset: 0

Peter Luschny, Sep 20 2014

			k\n
[1], 0, 1, 0,  1,   2,    9,   44,    265,      1854, ...  A000166
[2], 0, 1, 1,  3,  11,   53,   309,  2119,     16687, ...  A000255
[3], 0, 1, 2,  7,  32,  181,  1214,  9403,     82508, ...  A000153
[4], 0, 1, 3, 13,  71,  465,  3539,  30637,   296967, ...  A000261
[5], 0, 1, 4, 21, 134, 1001,  8544,  81901,   870274, ...  A001909
[6], 0, 1, 5, 31, 227, 1909, 18089, 190435,  2203319, ...  A001910
[7], 0, 1, 6, 43, 356, 3333, 34754, 398959,  4996032, ...  A176732
[8], 0, 1, 7, 57, 527, 5441, 61959, 770713, 10391023, ...  A176733
The referenced sequences may have a different offset or other small deviations.

Cf. A000166, A000255, A000153, A000261, A001909, A001910, A176732 - A176736.

A := (k,n) -> `if`(n<2,n,hypergeom([k,-n+1],[],1)*(-1)^(n+1));
seq(print(seq(round(evalf(A(k,n),100)), n=0..8)), k=1..8);

from mpmath import mp, hyp2f0
mp.dps = 25; mp.pretty = True
def A247490(k, n):
    if n < 2: return n
    if k == 1 and n == 2: return 0  # (failed to converge)
    return int((-1)^(n+1)*hyp2f0(k, -n+1, 1))
for k in (1..8): print([k], [A247490(k, n) for n in (0..8)])

A271698 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 2, 8, 1, 0, 0, 2, 28, 22, 1, 0, 0, 2, 72, 182, 52, 1, 0, 0, 2, 164, 974, 864, 114, 1, 0, 0, 2, 352, 4174, 8444, 3474, 240, 1, 0, 0, 2, 732, 15782, 61464, 57194, 12660, 494, 1, 0, 0, 2, 1496, 55286, 373940, 660842, 332528, 43358, 1004, 1, 0

Offset: 0

Peter Luschny, Apr 12 2016

			Triangle starts:
1,
1, 0,
0, 1, 0,
0, 2, 1, 0,
0, 2, 8, 1, 0,
0, 2, 28, 22, 1, 0,
0, 2, 72, 182, 52, 1, 0,
0, 2, 164, 974, 864, 114, 1, 0

Peter Luschny, Extensions of the binomial

A000255 (row sums), compare A028296 for alternating rows sums, A145654 and A005803 (diag. n,n-2).

Cf. A173018.

A271698 := (n,k) -> add(binomial(-j,-n)*combinat:-eulerian1(j,k), j=0..n):
seq(seq(A271698(n, k), k=0..n), n=0..10);

Mathematica
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A316666 Number of simple relaxed compacted binary trees of right height at most one with no sequences on level 1 and no final sequences on level 0.

Original entry on oeis.org

1, 0, 1, 3, 15, 87, 597, 4701, 41787, 413691, 4512993, 53779833, 695000919, 9680369943, 144560191149, 2303928046437, 39031251610227, 700394126116851, 13270625547477177, 264748979672169681, 5547121478845459983, 121784530649198053263, 2795749225338111831429, 66981491857058929294653

Offset: 0

Michael Wallner, Jul 10 2018

nonn

A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and at most n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. It is called simple if for nodes with two pointers both point to the same node. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. See the Wallner link.

a(n) is one of two "basis" sequences for sequences of the form a(0)=a, a(1)=b, a(n) = n*a(n-1) + (n-1)*a(n-2), the second basis sequence being A096654 (with 0 appended as a(0)). The sum of these sequences is listed as A000255. - Gary Detlefs, Dec 11 2018

G. C. Greubel, Table of n, a(n) for n = 0..448
Antoine Genitrini, Bernhard Gittenberger, Manuel Kauers and Michael Wallner, Asymptotic Enumeration of Compacted Binary Trees, arXiv:1703.10031 [math.CO], 2017.
Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017.

Cf. A000032, A000246, A001879, A051577, A213527, A288950, A288952, A288953 (subclasses of relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000166, A000255, A000262, A052852, A123023, A130905, A176408, A201203 (variants of simple relaxed compacted binary trees of right height at most one, see the Wallner link).
Cf. A000255, A096654.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (3*Exp(-x) + x-2)/(1-x)^2 )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Dec 12 2018

aseq := n-> 3*round((n+2)*n!/exp(1))-(n+2)*n!: bseq := n-> (n+2)*n!- 2* round((n+2)*n!/exp(1)): s := (a,b,n)-> a*aseq(n) + b*bseq( n): seq(s(1,0,n),n = 0..20);  # Gary Detlefs, Dec 11 2018

terms = 24;
CoefficientList[(3E^-z+z-2)/(1-z)^2 + O[z]^terms, z] Range[0, terms-1]! (* Jean-François Alcover, Sep 14 2018 *)

Vec(serlaplace((3*exp(-x + O(x^25)) + x - 2)/(1 - x)^2)) \\ Andrew Howroyd, Jul 10 2018

E.g.f.: (3*exp(-z)+z-2)/(1-z)^2.
a(n) ~ (3*exp(-1) - 1) * n * n!. - Vaclav Kotesovec, Jul 12 2018
a(n) = 3*round((n+2)*n!/e) - (n+2)*n!. - Gary Detlefs, Dec 11 2018
From Seiichi Manyama, Apr 25 2025: (Start)
a(n) = 3 * A000255(n) - n! - (n+1)!.
a(0) = 1, a(1) = 0; a(n) = n*a(n-1) + (n-1)*a(n-2). (End)

cp's OEIS Frontend

A350313 The Redstone permutation: a(1) = 2, a(2) = 1, otherwise the smallest number not occurring earlier which is strongly prime to n.

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A370383 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4].

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A370384 Number of permutations of [n] having no substring [k,k+1,k+2,k+3,k+4,k+5].

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A383380 Expansion of e.g.f. exp(-2*x) / (1-x)^4.

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A180185 Triangle read by rows: T(n,k) is the number of permutations of [n] having no 3-sequences and having k successions (0 <= k <= floor(n/2)); a succession of a permutation p is a position i such that p(i +1) - p(i) = 1.

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A195326 Numerators of fractions leading to e - 1/e (A174548).

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A218538 Triangle read by rows: T(n,k) is the number of permutations of{1,2,...,n} avoiding [x,x+1] having genus k (see first comment for definition of genus).

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A247490 Square array read by antidiagonals: A(k, n) = (-1)^(n+1)* hypergeom([k, -n+1], [], 1) for n>0 and A(k,0) = 0 (n>=0, k>=1).

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A271698 Triangle read by rows, T(n,k) = Sum_{j=0..n} C(-j,-n)*E1(j,k), E1 the Eulerian numbers A173018, for n>=0 and 0<=k<=n.

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