A317111 -id:A317111 - cp's OEIS frontend

A322282 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 8).

1, 1, 1, 1, 1, 1, 1, 1, 2, 18, 162, 1122, 6402, 31746, 141570, 580866, 2241096, 8693256, 43232904, 362491272, 4067218584, 45304757784, 459941563224, 4236342378840, 35804034476496, 281634733757520, 2106753678778320, 15739783039815120

Offset: 0

Seiichi Manyama, Dec 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..537
David Galvin, John Engbers, and Clifford Smyth, Reciprocals of thinned exponential series, arXiv:2303.14057 [math.CO], 2023.
Ira M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, arXiv:1807.09290 [math.CO], 2018.

Cf. A000142, A322251 (mod 3), A317111 (mod 4), A322276 (mod 5), A322262 (mod 6), A322297 (mod 7), A322298 (mod 9), A322283 (mod 10).

m = 28; CoefficientList[1/Normal[Exp[-x]+O[x]^8]+O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Feb 24 2019 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/sum(k=0, 7, (-x)^k/k!)))

E.g.f.: 1/(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - x^7/7!).

A322283 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 22, 242, 2002, 13442, 77506, 397826, 1862146, 8085506, 32978946, 127758774, 482490294, 2015041314, 13111486674, 144226353414, 1835958708870, 22030803357420, 240151251989220, 2389590181956120, 21944411982069720, 187919216043135720

Offset: 0

Seiichi Manyama, Dec 02 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..557
David Galvin, John Engbers, and Clifford Smyth, Reciprocals of thinned exponential series, arXiv:2303.14057 [math.CO], 2023.
Ira M. Gessel, Reciprocals of exponential polynomials and permutation enumeration, arXiv:1807.09290 [math.CO], 2018.

Cf. A000142, A322251 (mod 3), A317111 (mod 4), A322276 (mod 5), A322262 (mod 6), A322297 (mod 7), A322282 (mod 8), A322298 (mod 9).

m = 31; CoefficientList[1/Normal[Exp[-x]+O[x]^10]+O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, Feb 24 2019 *)

N=40; x='x+O('x^N); Vec(serlaplace(1/sum(k=0, 9, (-x)^k/k!)))

E.g.f.: 1/(1 - x + x^2/2! - x^3/3! + x^4/4! - x^5/5! + x^6/6! - x^7/7! + x^8/8! - x^9/9!).

A371605 Expansion of e.g.f. 1 / (1 - x + x^2/2 - x^3/3).

Original entry on oeis.org

1, 1, 1, 2, 10, 50, 230, 1260, 9240, 76440, 651000, 6006000, 62739600, 719518800, 8736327600, 112588476000, 1558917360000, 23070967920000, 360507459312000, 5925688056288000, 102619150714080000, 1869557514945120000, 35676360727207200000

Offset: 0

Ilya Gutkovskiy, Apr 01 2024

nonn

Cf. A006252, A009775, A217260, A317111, A355293, A371621.

nmax = 22; CoefficientList[Series[1/(1 - x + x^2/2 - x^3/3), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = a[1] = a[2] = 1; a[n_] := a[n] = n a[n - 1] - n (n - 1) a[n - 2]/2 + n (n - 1) (n - 2) a[n - 3]/3; Table[a[n], {n, 0, 22}]

a(n) = n * a(n-1) - n * (n-1) * a(n-2) / 2 + n * (n-1) * (n-2) * a(n-3) / 3.

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A322282 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 8).

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A322283 Number of permutations of [n] in which the length of every increasing run is 0 or 1 (mod 10).

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A371605 Expansion of e.g.f. 1 / (1 - x + x^2/2 - x^3/3).

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