A180189 -id:A180189 - cp's OEIS frontend

A190190 Positions of 1 in A190188; complement of A180189.

1, 2, 4, 5, 8, 12, 13, 15, 16, 18, 19, 24, 25, 26, 27, 29, 32, 35, 36, 37, 38, 40, 43, 47, 48, 50, 51, 54, 57, 59, 61, 62, 64, 65, 70, 71, 72, 73, 75, 76, 82, 83, 84, 86, 89, 93, 94, 95, 96, 97, 100, 103, 105, 106, 107, 108, 111, 114, 118, 119, 121, 122, 125, 128, 129, 130, 132, 133, 135, 140, 141, 142, 143, 144, 146, 151

Offset: 1

Clark Kimberling, May 05 2011

nonn

See A190188.

G. C. Greubel, Table of n, a(n) for n = 1..5549

Cf. A190188, A190189.

u = E; v = 1/E;
f[n_] := Floor[n*u + n*v] - Floor[n*u] - Floor[n*v]
t = Table[f[n], {n, 1, 120}] (* A190188 *)
Flatten[Position[t, 0]]      (* A190189 *)
Flatten[Position[t, 1]]      (* A190190 *)

A180188 Triangle read by rows: T(n,k) is the number of permutations of [n] with k circular successions (0<=k<=n-1). A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

Original entry on oeis.org

1, 0, 2, 3, 0, 3, 8, 12, 0, 4, 45, 40, 30, 0, 5, 264, 270, 120, 60, 0, 6, 1855, 1848, 945, 280, 105, 0, 7, 14832, 14840, 7392, 2520, 560, 168, 0, 8, 133497, 133488, 66780, 22176, 5670, 1008, 252, 0, 9, 1334960, 1334970, 667440, 222600, 55440, 11340, 1680, 360, 0

Offset: 1

Emeric Deutsch, Sep 06 2010

For example, p=(4,1,2,5,3) has 2 circular successions: (1,2) and (3,4).
Sum of entries in row n = n! = A000142(n).
T(n,0)=nd(n-1)=A000240(n).
T(n,1)=n(n-1)d(n-2)=A180189(n).
Sum(k*T(n,k), k>=0)=n! = A000142(n) if n>=2.

			T(3,2) = 3 because we have 123, 312, and 231.
The triangle starts:
1;
0,   2;
3,   0,  3;
8,  12,  0, 4;
45, 40, 30, 0, 5;

Alois P. Heinz, Rows n = 1..141, flattened
S. M. Tanny, Permutations and successions, J. Combinatorial Theory, Series A, 21 (1976), 196-202.

Cf. A000142, A000166, A000240, A180189.

A180188 := proc (n, k) n*binomial(n-1, k)*A000166(n-1-k) end proc:
for n to 10 do seq(A180188(n, k), k = 0 .. n-1) end do; # yields sequence in triangular form

T[n_, k_] := n*Binomial[n-1, k]*Subfactorial[n-1-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017 *)

T(n,k) = n*C(n-1,k)*d(n-1-k), where d(j) = A000166(j) are the derangement numbers (see Prop. 1 of the Tanny reference).

T(n,k) = n*A008290(n-1,k), 0<=k=1. - R. J. Mathar, Sep 08 2013

A182437 a(0)=0, a(n) = (a(n-1) XOR n) * (n+1).

Original entry on oeis.org

0, 2, 0, 12, 40, 270, 1848, 14840, 133488, 1334970, 14684560, 176214852, 2290793128, 32071103750, 481066556280, 7697064900464, 130850103307616, 2355301859537394, 44750735331210144, 895014706624203260, 18795308839108268040, 413496794460381897342

Offset: 0

Alex Ratushnyak, Apr 28 2012

for n<12, a(n)=A180189(n).

Cf. A182420, A180189.

nxt[{n_,a_}]:={n+1,BitXor[a,n+1](n+2)}; NestList[nxt,{0,0},30][[All,2]] (* Harvey P. Dale, Nov 01 2017 *)

a=0
for i in range(1,55):
  print(a, end=', ')
  a ^= i
  a *= i+1

a(0)=0, for n>0, a(n) = (a(n-1) XOR n) * (n+1).

A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

Original entry on oeis.org

0, 1, 2, 3, 20, 85, 534, 3703, 29672, 266985, 2669930, 29369131, 352429692, 4581585853, 64142202110, 962133031455, 15394128503504, 261700184559313, 4710603322067922, 89501463119290195, 1790029262385804260, 37590614510101889061, 826993519222241559782

Offset: 0

Ilya Gutkovskiy, Oct 11 2021

nonn

Cf. A000166, A000240, A030297, A053817, A180189, A212291.

Table[n! Sum[(-1)^k (k - 2)/(k - 1)!, {k, 1, n}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[x (1 + x) Exp[-x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

a(n) = n!*sum(k=1, n, (-1)^k * (k-2) / (k-1)!); \\ Michel Marcus, Oct 20 2021

E.g.f.: x * (1 + x) * exp(-x) / (1 - x).
a(0) = 0; a(n) = n * (a(n-1) + (-1)^n * (n-2)).
a(n) = n * (2 * A000166(n-1) + (-1)^n).

cp's OEIS Frontend

A190190 Positions of 1 in A190188; complement of A180189.

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A180188 Triangle read by rows: T(n,k) is the number of permutations of [n] with k circular successions (0<=k<=n-1). A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

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A182437 a(0)=0, a(n) = (a(n-1) XOR n) * (n+1).

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A348311 a(n) = n! * Sum_{k=1..n} (-1)^k * (k-2) / (k-1)!.

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