A000166 -id:A000166 - cp's OEIS frontend

A378162 Number k such that !k = A000166(k) is squarefree.

0, 2, 3, 6, 8, 11, 14, 15, 18, 20, 24, 27, 30, 32, 35, 36, 39, 42, 44, 47, 48, 54, 59, 60, 62, 63, 66, 68, 71, 72, 74, 75, 80, 83, 84, 86, 87, 90, 92, 95, 96, 98, 102, 104, 107, 108, 110, 114, 116

Offset: 1

Amiram Eldar, Nov 18 2024

0 and numbers k such that A378160(k) = A378161(k).
If k is a term, then k-1 is squarefree (since (k-1) | A000166(k)).
a(34) >= 83.

Cf. A000166, A005117, A378160, A378161.

Select[Range[0, 40], SquareFreeQ[Subfactorial[#]] &]

lista(kmax) = {my(s = 0); print1(0, ", "); for(k = 2, kmax, s = k * s + (-1)^k; if(issquarefree(s), print1(k, ", ")));}

a(34)-a(49) from Jinyuan Wang, Nov 25 2024

A065088 a(n) = A000166(n)*binomial(n,2).

Original entry on oeis.org

0, 0, 1, 6, 54, 440, 3975, 38934, 415324, 4805856, 60073245, 807651350, 11630179506, 178681848696, 2918470195459, 50511984152070, 923647710209400, 17795612550034304, 360361154138194809, 7652375096699313126, 170052779926651402990, 3947014523560698353400

Offset: 0

N. J. A. Sloane, Nov 10 2001

nonn

Robert Israel, Table of n, a(n) for n = 0..446

Cf. A000387.

with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(2):seq(count(ZLL, size=n)*binomial(n,2), n=0..21); # Zerinvary Lajos, Jun 11 2008

Table[Subfactorial[n]*Binomial[n, 2], {n, 0, 22}] (* Zerinvary Lajos, Jul 09 2009 *)

A109742 a(n) = d(n-1) + d(n-2) + (n-1)[d(n-2) + 2d(n-3) + d(n-4)], where d(n), the derangement numbers, are given in A000166. (Let d(n) = 0 if n < 0.)

Original entry on oeis.org

1, 2, 5, 9, 27, 123, 693, 4653, 36111, 317583, 3118617, 33804177, 400755267, 5156954019, 71572594557, 1065571143093, 16938122939703, 286298719063863, 5127206924693601, 96975312507734553, 1931609062232400747, 40414621201681598667, 886153986344092389957

Offset: 1

N. J. A. Sloane, Aug 13 2005

nonn

Y.-R. Liu and M. R. Murthy, Sieve methods in combinatorics, J. Combinatorial Theory, Ser. A, 111 (2005), 1-23.

Cf. A000166, A109743.

a[1] = 1; a[2] = 2; a[3] = 5; a[n_] := a[n] = ((2*n^3 - 27*n^2 + 115*n - 150)*a[n - 3] + (2*n^3 - 23*n^2 + 85*n - 100)*a[n - 2] + 3*(2*n - 9)*a[n - 1])/(2*n - 9); Table[a[n], {n, 1, 23}]
(* or: *)
d[n_] := If[n < 0, 0, Subfactorial[n]]; Table[(n - 1)*(d[n - 4] + 2*d[n - 3] + d[n - 2]) + d[n - 2] + d[n - 1], {n, 1, 23}](* Jean-François Alcover, Nov 03 2016 *)

d(n)=if(n>0, n!\/exp(1), n==0)
a(n)=d(n-1) + d(n-2) + (n-1)*(d(n-2) + 2*d(n-3) + d(n-4)) \\ Charles R Greathouse IV, Nov 03 2016

A131631 Supersubfactorials: partial product of positive subfactorials (A000166).

Original entry on oeis.org

1, 2, 18, 792, 209880, 389117520, 5771780174160, 770509566129663360, 1028600220910021528728960, 15104551945968674840127424147200, 2661646219535110627933754465838408595200

Offset: 2

Jonathan Vos Post, Sep 01 2007

This is to subfactorials (A000166, rencontres numbers, or derangements) as superfactorials (A000178) are to factorials (A000142).

			a(2) = 1.
a(3) = 1 * 2 = 2.
a(4) = 1 * 2 * 9 = 18 = 2 * 3^2.
a(5) = 1 * 2 * 9 * 44 = 792 = 2^3 * 3^2 * 11.
a(6) = 1 * 2 * 9 * 44 * 265 = 209880 = 2^3 * 3^2 * 5 * 11 * 53.
a(7) = 1 * 2 * 9 * 44 * 265 * 1854 = 389117520 = 2^4 * 3^4 * 5 * 11 * 53 * 103.

Cf. A000142, A000166, A000178.

Table[Product[k!*Sum[(-1)^j/j!, {j,0,k}], {k,2,n}], {n,2,15}] (* Vaclav Kotesovec, Jul 11 2015 *)

a(n) = Product_{k=2..n} A000166(k).

a(n) ~ c * n^(n^2/2 + n + 5/12) * (2*Pi)^((n+1)/2) / (A * exp(3*n^2/4 + 2*n - 13/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and c = 1.2517384488693662195086340541087053383189277225386098721341690164735... . - Vaclav Kotesovec, Jul 11 2015

A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 4, 0, 3, 0, 27, 0, 8, 96, 0, 256, 0, 45, 640, 2430, 0, 3125, 0, 264, 8640, 29160, 61440, 0, 46656, 0, 1855, 118272, 688905, 1146880, 1640625, 0, 823543, 0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248, 0, 16777216, 0, 133497, 34172928, 438143580, 1453326336, 2214843750, 1693052928, 1452729852, 0, 387420489

Offset: 0

Roger L. Bagula, Feb 15 2009

			Triangle begins as:
  1;
  0,     1;
  0,     0,       4;
  0,     3,       0,       27;
  0,     8,      96,        0,      256;
  0,    45,     640,     2430,        0,     3125;
  0,   264,    8640,    29160,    61440,        0,    46656;
  0,  1855,  118272,   688905,  1146880,  1640625,        0, 823543;
  0, 14832, 1899520, 16166304, 41287680, 43750000, 47029248,      0, 16777216;

J. Riordan, Combinatorial Identities, Wiley, 1968, p.194.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000166, A000240, A000312, A137341.

A000166[n_]:= A000166[n]= If[n==0, 1, n*A000166[n-1] + (-1)^n];
T[n_, k_]:= If[n==0, 1, Binomial[n, k]*A000166[n-k]*k^n];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 10 2021 *)

def A000166(n): return 1 if (n==0) else n*A000166(n-1) + (-1)^n
def A156788(n,k): return 1 if (n==0) else binomial(n,k)*k^n*A000166(n-k)
flatten([[A156788(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1.
T(n, k) = binomial(n, k)*b(n-k)*k^n, where b(n) = n*b(n-1) + (-1)^n and b(0) = 1.
Sum_{k=0..n} T(n, k) = A137341(n).
From G. C. Greubel, Jun 10 2021: (Start)
T(n, 1) = A000240(n).
T(n, n) = A000312(n). (End)

Edited by G. C. Greubel, Jun 10 2021

A260656 a(n) = lcm{!2, !3, ..., !n}, where !n = A000166(n) is subfactorial.

Original entry on oeis.org

1, 2, 18, 396, 104940, 10808820, 160327227060, 486432806900040, 72152091814676033160, 105952244289903723626034120, 1697305261921685687642685992397720, 108004858262683508632706244802225075247640, 266448824855803491635798907952730108331437779905720

Offset: 2

Vladimir Reshetnikov, Nov 13 2015

nonn

a(n) <= A131631(n).

			For n = 5, a(5) = lcm(!2, !3, !4, !5) = lcm(1, 2, 9, 44) = 396.

Alois P. Heinz, Table of n, a(n) for n = 2..45
Index entries for sequences related to lcm's

Cf. A000166, A131631.

b:= proc(n) option remember;
     `if`(n=0, 1, n*b(n-1)+(-1)^n)
    end:
a:= n-> ilcm(seq(b(i), i=2..n)):
seq(a(n), n=2..15);  # Alois P. Heinz, May 08 2020

LCM@@@Subfactorial@Range[2, Range[2, 14]]

a(n) = lcm(vector(n-1, k, if(k+1,round((k+1)!/exp(1)),1))); \\ Altug Alkan, Nov 13 2015

A272988 Convolution of the sequence of derangement numbers A000166 with itself.

Original entry on oeis.org

1, 0, 2, 4, 19, 92, 552, 3832, 30453, 272552, 2713710, 29752156, 356133959, 4620985700, 64600445812, 967927029168, 15473320537001, 262864036323600, 4728905854617562, 89808092596277364, 1795480569403712699, 37693097921348983852, 829024574048725950016, 19063166411687276701736

Offset: 0

J. C. George, May 12 2016

			For n = 4, we get 1*9 + 0*2 + 1*1 + 2*0 + 9*1 = 19.

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Cf. A000166.

Table[Sum[Subfactorial[k] Subfactorial[n - k], {k, 0, n}], {n, 0, 30}] (* Emanuele Munarini, Oct 06 2017 *)

a(n) = Sum_{i=0..n} A000166(i)*A000166(n-i).
G.f.: ( 1/(1 + x) + Sum_{k>=1} k^k*x^k/(1 + (k + 1)*x)^(k+1) )^2. - Ilya Gutkovskiy, Apr 13 2017
a(n) ~ 2*exp(-1)*n!. - Vaclav Kotesovec, Apr 13 2017

A281682 Decimal expansion of Sum_{n>=2} 1/A000166(n).

Original entry on oeis.org

1, 6, 3, 8, 2, 2, 7, 0, 7, 4, 5, 0, 5, 3, 7, 0, 6, 4, 7, 5, 4, 2, 8, 9, 3, 1, 1, 4, 1, 5, 1, 1, 2, 2, 6, 6, 1, 0, 6, 3, 5, 9, 3, 2, 4, 9, 6, 4, 4, 4, 3, 6, 1, 6, 4, 7, 2, 3, 2, 6, 2, 8, 2, 8, 7, 2, 6, 3, 0, 5, 8, 2, 9, 4, 4, 0, 6, 8, 2, 2, 3, 9, 8, 1, 8, 3, 0, 3, 9, 5, 6, 7, 2, 0, 7, 3, 2, 9, 9, 6, 0, 9, 1, 0, 8, 1, 3, 9, 0, 9, 1, 5, 3

Offset: 1

Miguel Alejandro Moreno Barrientos, Jan 26 2017

			1.63822707450537064754289311415112266106359324964443616472326282872630582...

Wikipedia, Subfactorial/Derangement definition

Cf. A000166.

PrecisionDigits ≔ 1000
NotationDigits ≔ 1000
sum(1/ROUND(n!/e), n, 2, 500)

a[n_]:=If[n>0, Round[n!/E], 1]; RealDigits[Sum[1/a[n], {n, 2, 500}], 10, 113][[1]](* Indranil Ghosh, Mar 12 2017 *)

Equals Sum_{n>=2} 1/round(n!/e).

A295165 Numbers n such that !n and n!! (A000166(n) and A006882(n)) are coprime.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 12, 14, 17, 18, 20, 24, 30, 32, 33, 44, 48, 54, 62, 65, 68, 72, 74, 80, 84, 98, 102, 110, 114, 128, 140, 150, 158, 168, 180, 182, 198, 200, 212, 224, 228, 230, 234, 252, 257, 264, 270, 272, 278, 282, 308, 312, 314, 318, 332, 348, 354, 374, 380, 384, 402, 410, 420, 422, 432

Offset: 1

Robert Israel, Nov 16 2017

nonn

Odd n is in the sequence iff !n is not divisible by any odd primes < n.
Even n is in the sequence iff !n is not divisible by any odd primes < n/2.
All odd terms are in A083318, all even terms > 2 are in A008864, but both of these are strict inclusions.
Odd terms include 1,3,5,9,17,33,65,257,513,32769.

			!5 = 44 and 5!! = 15 are coprime so 5 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000166, A006882, A008864, A083318.

sf:= proc(n) option remember; n*procname(n-1)+(-1)^n end proc:
sf(0):= 1:
select(n -> igcd(sf(n),doublefactorial(n))=1, [$0..1000]);

Select[Range[0, 1000], CoprimeQ[Subfactorial[#], #!!]&] (* Jean-François Alcover, Oct 16 2020 *)

A371998 a(n) = A000166(floor(n/2)) if n is even otherwise A000240(floor((n + 1)/2)).

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 2, 8, 9, 45, 44, 264, 265, 1855, 1854, 14832, 14833, 133497, 133496, 1334960, 1334961, 14684571, 14684570, 176214840, 176214841, 2290792933, 2290792932, 32071101048, 32071101049, 481066515735, 481066515734, 7697064251744, 7697064251745

Offset: 0

Peter Luschny, Apr 25 2024

nonn

Cf. A000166, A000240.

a := n -> if irem(n,2) = 0 then A000166(iquo(n,2)) else A000240(iquo(n+1,2)) fi:
seq(a(n), n = 0..32);

from functools import cache
@cache
def sf(n):
    if n == 0: return 1
    return n * sf(n - 1) + (-1 if n % 2 else 1)
def a(n):
    h, r = divmod(n, 2)
    return sf(h) * (h + 1) if r else sf(h)
print([a(n) for n in range(33)])

cp's OEIS Frontend

A378162 Number k such that !k = A000166(k) is squarefree.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Extensions

A065088 a(n) = A000166(n)*binomial(n,2).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A109742 a(n) = d(n-1) + d(n-2) + (n-1)[d(n-2) + 2d(n-3) + d(n-4)], where d(n), the derangement numbers, are given in A000166. (Let d(n) = 0 if n < 0.)

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A131631 Supersubfactorials: partial product of positive subfactorials (A000166).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A156788 Triangle T(n, k) = binomial(n, k)*A000166(n-k)*k^n with T(0, 0) = 1, read by rows.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

Sage

Formula

Extensions

A260656 a(n) = lcm{!2, !3, ..., !n}, where !n = A000166(n) is subfactorial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A272988 Convolution of the sequence of derangement numbers A000166 with itself.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A281682 Decimal expansion of Sum_{n>=2} 1/A000166(n).

Views

Author

Keywords

Examples

Links

A156788 Triangle T(n, k) = binomial(n, k)A000166(n-k)k^n with T(0, 0) = 1, read by rows.