A265165 -id:A265165 - cp's OEIS frontend

A306699 Periods of A265165(k) mod n.

2, 12, 8, 1, 12, 84, 8, 36, 2, 1, 24, 104, 84, 12, 16, 544, 36, 1, 8, 84, 2, 1012, 24, 1, 104, 108, 168, 1, 12, 1, 32, 12, 544, 84, 72, 2664, 2, 312, 8, 1, 84, 3612, 8, 36, 1012, 4324, 48, 588, 2, 1632, 104, 5512, 108, 1, 168, 12, 2, 1, 24, 1, 2, 252, 64, 104, 12, 2948, 544, 3036, 84, 1, 72, 10512, 2664

Offset: 2

Cyril Banderier, Mar 05 2019

nonn

Let b(k) be the sequence A265165(k).
a(n) = period({b(k) mod n}) = smallest p > 0 such that b(k+p) = b(k) mod n (for all large enough k).
The sequences b(k) and a(n) were introduced in the Banderier-Baril-Moreira article, they have many noteworthy arithmetical properties (proven in the Banderier-Luca article).

			A265165(k) mod 15 = (10,5,10,10,0,10,5,10,5,5,0,5)... and this pattern of length 12 repeats, therefore a(15) = 12.

Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, DMTCS 18:2#12, p. 1-17, 2017.
Cyril Banderier, Florian Luca, On the period mod m of polynomially-recursive sequences: a case study, arXiv:1903.01986 [math.NT], 2019.

Cf. A265163, A265164, A265165.

The Banderier-Luca article proves the following properties:
a(n) = 1 iff n is a product of primes in 0,1,4 mod 5.
a(n) = 2 iff n/2 is a product of primes in 0,1,4 mod 5.
If a(n) is not 1, then it is an even number.
For any prime p, a(p) | 2 p (p-1).
For any prime p not in 0,1,4 mod 5, (and p^r <> 4), a(p^r) = p^r a(p).
a(n) is an "lcm-multiplicative" sequence: a(n1*n2) = lcm(a(n1), a(n2)) (for n1,n2 coprime), this implies that if n = p1^e1 ... pk^ek (factorization in distinct primes) then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

A265164 Sum of the n-th row of the array A265163(n, k).

Original entry on oeis.org

1, 3, 15, 101, 841, 8283, 93815, 1198029, 16997041, 264864419, 4492081151, 82299283669, 1618674299769, 33997164987019, 759059595497511, 17945237236457533, 447676430154815137, 11748882878147100691, 323494584038834863087, 9322205037165367256837

Offset: 0

Cyril Banderier, Dec 07 2015; revised Feb 06 2017

nonn

A right-jump in a permutation consists of taking an element and moving it somewhere to its right.
The set P(k) of permutations reachable from the identity after at most k right-jumps is a permutation-pattern avoiding set: it coincides with the set of permutation avoiding a set of patterns.
We define B(k) to be the smallest such set of "forbidden patterns" (the permutation pattern community calls such a set a "basis" for P(k), and its elements can be referred to as "right-jump basis permutations").
The number b(n,k) of permutations of size n in B(k) is given by the array A265163.
The row sums give the present sequence (i.e. this counts the permutations of any size in the basis B(k)).
The column sums give the sequence A265165 (i.e. this counts the permutations of size n in any B(k)).

			G.f. = 1 + 3*x + 15*x^2 + 101*x^3 + 841*x^4 + 8283*x^5 + 93815*x^6 + 1198029*x^7 + ...
The basis permutations for B(1) are 312, 321, and 2143, thus a(1)=3.
The basis permutations for B(2) are 4123, 4132, 4213, 4231, 4312, 4321, 21534, 21543, 31254, 32154, 31524, 31542, 32514, 32541, and 214365, thus a(2)=15.

Vaclav Kotesovec, Table of n, a(n) for n = 0..200
Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, Permutation Patterns 2015.

Cf. A265163, A265165.

a[ n_] := Module[ {A, s, F}, If[ n < 0, 0, A = 1 - x + O[x]^(2 n + 3); s = Sqrt[1 + 4 y + O[y]^(n + 2)]; F = y ((1 - 1/s) A^((1 + s)/2) + (1 + 1/s) A^((1 - s)/2))/2; Sum[ SeriesCoefficient[ SeriesCoefficient[ F, {x, 0, n + k}] (n + k)!, {y, 0, k}], {k, 2, 2 + n}]]]; (* Michael Somos, Jan 27 2017 *)

{a(n) = my(A, s, F); if( n<0, 0, A = 1 - x + x * O(x^(2*n+2)); s = sqrt(1 + 4*y + y * O(y^(n+1))); F = y * ((1 - 1/s) * A^((1 + s)/2) + (1 + 1/s) * A^((1 - s)/2)) / 2; sum(k=2, 2+n, polcoeff( polcoeff( F, n+k) * (n+k)!, k)))}; /* Michael Somos, Jan 27 2017 */

A265163 Array of basis permutations, seen as a triangle read by rows: Row k (k >= 0) gives the values of b(n, k) = number of permutations of size n (2 <= n <= 2(k+1)) in the permutation basis B(k) (see Comments for further details).

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 6, 8, 1, 0, 0, 0, 24, 58, 18, 1, 0, 0, 0, 0, 120, 444, 244, 32, 1, 0, 0, 0, 0, 0, 720, 3708, 3104, 700, 50, 1, 0, 0, 0, 0, 0, 0, 5040, 33984, 39708, 13400, 1610, 72, 1, 0, 0, 0, 0, 0, 0, 0, 40320, 341136, 525240, 244708, 43320, 3206, 98, 1

Offset: 0

Cyril Banderier, Dec 07 2015, with additional comments added Feb 06 2017

A right-jump in a permutation consists of taking an element and moving it somewhere to its right.
The set P(k) of permutations reachable from the identity after at most k right-jumps is a permutation-pattern avoiding set: it coincides with the set of permutation avoiding a set of patterns.
We define B(k) to be the smallest such set of "forbidden patterns" (the permutation pattern community calls such a set a "basis" for P(k), and its elements can be referred to as "right-jump basis permutations").
The number b(n,k) of permutations of size n in B(k) is given by the array in the present sequence.
The row sums give the sequence A265164 (i.e. this counts the permutations of any size in the basis B(k)).
The column sums give the sequence A265165 (i.e. this counts the permutations of size n in any B(k)).

			The number b(n, k) of basis permutations of length n where 2<=n<=11.
k\n |  2  3  4   5   6    7    8     9     10      11  |  #B_k
0   |  1                                               |     1
1   |  0  2  1                                         |     3
2   |  0  0  6   8   1                                 |    15
3   |  0  0  0  24  58   18    1                       |   101
4   |  0  0  0   0 120  444  244    32      1          |   841
5   |  0  0  0   0   0  720 3708  3104    700      50  |  8232
6   |  0  0  0   0   0    0 5040 33984  39708   13400  | 78732
----+--------------------------------------------------+------
Sum |  1  2  7  32 179 1182 8993 77440 744425 7901410  |
----+--------------------------------------------------+------

Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, Permutation Patterns 2015.

Cf. A265164 (row sums B(k)), A265165 (column sums).

A323620 Expansion of e.g.f. 2sqrt(1 + x)sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

Original entry on oeis.org

0, 1, 0, 1, -4, 19, -108, 719, -5496, 47465, -457160, 4858865, -56490060, 713165035, -9715762980, 142069257055, -2219386098160, 36889108220305, -650018185589520, 12103669982341025, -237476572759473300, 4896758300881695875, -105866710959427454300, 2394660132226522508975

Offset: 0

Ilya Gutkovskiy, Jan 20 2019

sign

G. C. Greubel, Table of n, a(n) for n = 0..449

Cf. A000045, A048994, A178840, A213593, A263576, A265165.

[(&+[StirlingFirst(n,k)*Fibonacci(k): k in [0..n]]): n in [0..25]]; // G. C. Greubel, Feb 07 2019

FullSimplify[nmax = 23; CoefficientList[Series[2 Sqrt[1 + x] Sinh[Sqrt[5] Log[1 + x]/2]/Sqrt[5], {x, 0, nmax}], x] Range[0, nmax]!]
Table[Sum[StirlingS1[n, k] Fibonacci[k], {k, 0, n}], {n, 0, 23}]

{a(n) = sum(k=0,n, stirling(n,k,1)*fibonacci(k))};
vector(25, n, n--; a(n)) \\ G. C. Greubel, Feb 07 2019

[sum((-1)^(k+n)*stirling_number1(n,k)*fibonacci(k) for k in (0..n)) for n in (0..25)] # G. C. Greubel, Feb 07 2019

a(n) = Sum_{k=0..n} Stirling1(n,k)*A000045(k).
From Vaclav Kotesovec, Jan 21 2019: (Start)
a(n) = -(-1)^n * cos(sqrt(5)*Pi/2) * (Gamma((3 + sqrt(5))/2) * Gamma(n - (1 + sqrt(5))/2) - Gamma((3 - sqrt(5))/2) * Gamma(n + (sqrt(5) - 1)/2)) / (Pi*sqrt(5)).
a(n) ~ -(-1)^n * n! / (sqrt(5) * Gamma((sqrt(5)-1)/2) * n^((3 - sqrt(5))/2)).
a(n) = -2*(n-2)*a(n-1) - (n^2 - 5*n + 5)*a(n-2). (End)

A354018 Expansion of e.g.f. -log(1-x)/(1 + log(1-x) - log(1-x)^2).

Original entry on oeis.org

0, 1, 3, 20, 172, 1864, 24248, 368136, 6388128, 124711944, 2705241672, 64550432352, 1680280323984, 47383464508080, 1438986494794704, 46821994627363968, 1625069178022566528, 59927028756823323648, 2339899614887520358656, 96439023491479275172608

Offset: 0

Seiichi Manyama, May 14 2022

nonn

Cf. A000045, A005444, A005445, A265165, A320352, A354013.

Table[Sum[k! * Fibonacci[k] * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 15 2022 *)
With[{nn=20},CoefficientList[Series[-Log[1-x]/(1+Log[1-x]-Log[1-x]^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 22 2024 *)

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-x)/(1+log(1-x)-log(1-x)^2))))

a(n) = sum(k=0, n, k!*fibonacci(k)*abs(stirling(n, k, 1)));

a(n) = Sum_{k=0..n} k! * Fibonacci(k) * |Stirling1(n,k)|.

a(n) ~ n! * (sqrt(5) - 1) / (2 * sqrt(5) * exp((sqrt(5) - 1)/2) * (1 - exp((1 - sqrt(5))/2))^(n+1)). - Vaclav Kotesovec, May 15 2022

cp's OEIS Frontend

A306699 Periods of A265165(k) mod n.

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A265164 Sum of the n-th row of the array A265163(n, k).

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A265163 Array of basis permutations, seen as a triangle read by rows: Row k (k >= 0) gives the values of b(n, k) = number of permutations of size n (2 <= n <= 2(k+1)) in the permutation basis B(k) (see Comments for further details).

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A323620 Expansion of e.g.f. 2*sqrt(1 + x)*sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).

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A354018 Expansion of e.g.f. -log(1-x)/(1 + log(1-x) - log(1-x)^2).

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Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A323620 Expansion of e.g.f. 2sqrt(1 + x)sinh(sqrt(5)*log(1 + x)/2)/sqrt(5).