A001563 -id:A001563 - cp's OEIS frontend

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1

Offset: 1

Alois P. Heinz, Feb 18 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    2,    1                        ;
  :                     1,    4,    6,    4,    1                  ;
  :               1,    6,   18,   24,   18,    6,   1             ;
  :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
  :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
  :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

A061018 Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 10, 13, 15, 24, 42, 56, 67, 76, 120, 216, 294, 358, 411, 455, 720, 1320, 1824, 2250, 2612, 2921, 3186, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 40320, 75600, 106560, 133800, 157824, 179058, 197864, 214551, 229384

Offset: 1

Wouter Meeussen, May 23 2001

Row sums of n are the number of derangements (permutations without fixed point) of n+1, i.e. A000166(n+1).

			For n=3, the permutations are (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1); and (x, 2, 3), (x, 3, 2) have a fixed point x in position 1, (x, x, 3), (x, 3, 2), (3, x, 1) have a fixed point x in positions 1 or 2 and (x, x, x), (2, 1, x), (x, 3, 2), (3, x, 1) have a fixed point x in positions 1, 2 or 3, hence {2, 3, 4}
{1},
{1, 1},
{2, 3, 4},
{6, 10, 13, 15},
{24, 42, 56, 67, 76},
{120, 216, 294, 358, 411, 455},
{720, 1320, 1824, 2250, 2612, 2921, 3186}, ...

Columns: A000142, A007680, A002467, A180191.

A061018 := proc(n,m): (n-1)! + add(A061312(n-2,k), k=0..m-2) end: A061312:= proc(n,m): if m=-1 then 0 elif m=0 then n*n! else procname(n,m-1) - procname(n-1,m-1) fi: end: seq(seq(A061018(n,m), m=1..n), n=1..8); # Johannes W. Meijer, Jul 27 2011
T := (n, k) -> `if`(n=k,n!-GAMMA(n+1,-1)/exp(1),n!*(1-hypergeom([-k],[-n],-1))):
for n from 1 to 9 do seq(simplify(T(n,k)), k=1..n) od; # Peter Luschny, Oct 03 2017

Table[Count[Permutations[Range[n]], p_/;( Times@@Take[(p-Range[n]), k]===0)], {n, 7}, {k, n}]

a(n,m) = (n-1)! + Sum_{k=0..m-2} T(n-2, k) where T(n,-1) = 0, T(0,0) = 0, T(n,0) = A001563(n) = n*n!, T(n,m) = T(n,m-1) - T(n-1,m-1) (see A061312).

T(n, k) = n!*(1 - hypergeom([-k], [-n], -1)) for 1 <= k < n and T(n, n) = n! -Gamma(n+1, -1)/exp(1). - Peter Luschny, Oct 03 2017

Edited and information added by Johannes W. Meijer, Jul 27 2011

A069149 Numbers k such that k*k!/A062758(k) is an integer where A062758(k) is the product of squares of divisors of k.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 12, 13, 15, 17, 19, 21, 23, 24, 25, 27, 29, 30, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 69, 70, 71, 72, 73, 75, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 90, 91, 93, 95, 96, 97, 98, 99

Offset: 1

Benoit Cloitre, Apr 08 2002

Also numbers k such that k! is divisible by k ^ (tau(k) - 1). - David A. Corneth, Apr 23 2018

Ivan Neretin, Table of n, a(n) for n = 1..10000

Complement of A069148.

Cf. A001563 (n*n!), A062758.

Select[Range[100], Divisible[#!*#, Times @@ Divisors[#]^2] &] (* Ivan Neretin, Apr 22 2018 *)
Select[Range[100], Divisible[#!, #^(DivisorSigma[0, #] - 1)] &] (* Amiram Eldar, Jul 07 2022 *)

for(n=1, 320, if((n*(n!))%(n^numdiv(n))==0, print1(n, ", ")))

is(n) = {my(f = factor(n), qdiv = numdiv(f)); for(i = 1, #f~, cn = n; t = 0; while(cn \= f[i,1], t += cn); if(t < f[i,2] * (qdiv - 1), return(0))); 1} \\ David A. Corneth, Apr 27 2018

A094304 Sum of all possible sums formed from all but one of the previous terms, starting 1.

Original entry on oeis.org

1, 0, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000

Offset: 1

Amarnath Murthy, Apr 29 2004

nonn

Apart from initial 1, same sequence as A001563. Additive analog of A057438.

a(1) = 1, for n >= 2: a(n) = sum of previous terms * (n-2) = (Sum_(i=1...n-2) a(i)) * (n-2). a(n) = A001563(n-2) = A094258(n-1) for n >= 3. - Jaroslav Krizek, Oct 16 2009

			a(2) = 0 as there is only one previous term and empty sum is taken to be 0.
a(4) = (a(1) +a(2))+ (a(1) +a(3)) + (a(2) +a(3)) = (1+0) +(1+1) +(0+1) = 4.
a(5) = (a(1)+a(2)+a(3)) +(a(1)+a(2)+ a(4)) +(a(1)+a(3)+a(4)) +(a(2)+a(3)+a(4)) = (1+0+1) +(1+0+4) +(1+1+4) +(0+1+4) = 2 + 5 + 6 + 5 = 18.

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A001563, A057438, A122974.

a := n -> (n-2)*(n-2)!: 1,seq(a(n), n=2..23); # Emeric Deutsch, May 01 2008

In[2]:= l = {1}; Do[k = Length[l] - 1; p = Plus @@ Flatten[Select[Subsets[l], Length[ # ]==k& ]]; AppendTo[l, p], {n, 20}]; l (* Ryan Propper, May 28 2006 *)

v=vector(30);v[1]=1;v[2]=0;for(n=3,#v,s=0;for(i=1,2^(n-1)-1, vb=binary(i); if(hammingweight(vb)==n-2,s=s+sum(j=1,#vb, if(vb[j], v[n-#vb+j-1]))));v[n]=s;print1(s,",")) /* Ralf Stephan, Sep 22 2013 */

a(n) = (n-2)!(n-2) for n>=2. - Emeric Deutsch, May 01 2008
G.f.: x*T(0), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 -x -2*x*k)*(1 -3*x -2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2013
a(n) = S1(n,1) - S1(n-1,1), where S1 are the unsigned Stirling cycle numbers. - Peter Luschny, Apr 10 2016
a(n) = A122974(n-1,n-1). - Alois P. Heinz, Nov 24 2019

Edited by N. J. A. Sloane, May 29 2006

A145572 Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.

Original entry on oeis.org

1, 3, 49, 12845057, 1017690263500988729456314874071089153, 4222921592695952872362526736376161058920018764920519780147745963811744865992371113095993596088044297100172572224585271942341064532181870606866447799704872724575357044373908131956500952542608981420222196042850818326529

Offset: 1

Wolfdieter Lang Mar 06 2009

a(n) is A145571(n) (a decimal number with digits only from {0,1}) read as base 2 number converted back into decimal notation.
The sequence of digit lengths is 1,1,2,8,37,217,1518,... (see A317873).
This sequence gives the numerators of the partial sums for the constant A092874 (called there "binary" Liouville number). See the B(n) formula below. - Wolfdieter Lang, Apr 10 2024

			a(3)=49, because A145571(3)=110001, and the binary number 110001 translates to 2^5+2^4+2^0=32+16+1 = 49.

Cf. A001563, A092874, A145571 (numerators of approximations for Liouville's number).

Cf. A317873.

a[n_] := FromDigits[RealDigits[Sum[1/10^k!, {k, n}], 10, n!][[1]], 2]; Array[a, 6] (* Robert G. Wilson v, Aug 08 2018 *)
Block[{k = 0}, NestList[#*2^(++k*k!) + 1 &, 1, 5]] (* Paolo Xausa, Jun 27 2024 *)

a(n) = A145571(n) interpreted as number in binary notation, then converted to decimal notation.
From Wolfdieter Lang, Apr 10 2024: (Start)
a(n) = Sum_{j=0..n} 2^(n! - j!) = 2^(n!)*B(n) = numerator(B(n)), where B(n) := Sum_{j=1..n} 1/2^(j!), for n >= 1 (Proof from the positions of 1 in A145571).
a(1) = 1, and a(n) = a(n-1)*2^z(n) + 1, where z(n) = n! - (n-1)! = A001563(n-1), for n >= 2.
(End)

A154306 a(n) = (n+1)^3*(3+n)!/6.

Original entry on oeis.org

1, 32, 540, 7680, 105000, 1451520, 20744640, 309657600, 4849891200, 79833600000, 1381360780800, 25107347865600, 478826764416000, 9568689242112000, 200074178304000000, 4370687116443648000, 99607063051431936000

Offset: 0

Omar E. Pol, Jan 06 2009

Row 3 of square array A152818.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000142, A001563, A055533, A152818, A154307, A154308.

[(n+1)^3*Factorial(3+n)/6: n in [0..20]]; // Vincenzo Librandi, Mar 27 2012

Table[(n+1)^3*(3+n)!/6,{n,0,20}] (* Vincenzo Librandi, Mar 27 2012 *)

a(n) = (n+1)^3*(n+3)!/6 \\ Charles R Greathouse IV, Sep 10 2016

E.g.f.: (1 + 25*x + 67*x^2 + 27*x^3)/(1-x)^7. - R. J. Mathar, Dec 21 2011

More terms from Sean A. Irvine, Dec 01 2009

A154307 a(n) = (n+1)^4*(4+n)!/24.

Original entry on oeis.org

1, 80, 2430, 53760, 1050000, 19595520, 363031200, 6812467200, 130947062400, 2594592000000, 53182390060800, 1129830653952000, 24898991749632000, 569337009905664000, 13505007035520000000, 332172220849717248000

Offset: 0

Omar E. Pol, Jan 06 2009

nonn

Row 4 of square array A152818.

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000142, A001563, A055533, A152818, A154306, A154308.

[(n+1)^4*Factorial(4+n)/24: n in [0..20]]; // Vincenzo Librandi, Sep 11 2016

Table[(n + 1)^4*(4 + n)!/24, {n, 0, 25}] (* G. C. Greubel, Sep 10 2016 *)

E.g.f.: (1 + 71*x + 531*x^2 + 821*x^3 + 256*x^4)/(1-x)^9. - R. J. Mathar, Dec 21 2011

Extended by Max Alekseyev, Apr 13 2009

A154308 a(n) = (n+1)^5*(5+n)!/120.

Original entry on oeis.org

1, 192, 10206, 344064, 9450000, 235146240, 5590680480, 130799370240, 3064161260160, 72648576000000, 1755018872006400, 43385497111756800, 1100535435333734400, 28694585299245465600, 769785401024640000000, 21259022134381903872000, 604515265659140419584000, 17698965059877321572352000

Offset: 0

Omar E. Pol, Jan 06 2009

nonn

Row 5 of square array A152818.

G. C. Greubel, Table of n, a(n) for n = 0..250
Clark Kimberling, Polynomials associated with reciprocation, JIS 12 (2009) 09.3.4 section 6.

Cf. A000142, A001563, A055533, A152818, A154306, A154307.

[(n+1)^5*Factorial(5+n)/120: n in [0..20]]; // Vincenzo Librandi, Sep 11 2016

Table[(n + 1)^5*(5 + n)!/120, {n, 0, 25}] (* G. C. Greubel, Sep 10 2016 *)

for(n=0,25, print1((n+1)^5*(5+n)!/120, ", ")) \\ G. C. Greubel, Nov 24 2017

a(n) = A000142(n+1)*A000583(n+1)*A000389(n+5). - R. J. Mathar, Jan 17 2009

E.g.f.: (1 + 181*x + 3046*x^2 + 11606*x^3 + 12281*x^4 + 3125*x^5)/(1-x)^11. - R. J. Mathar, Dec 21 2011

More terms from R. J. Mathar, Jan 17 2009

A161158 a(n) = A003696(n+1)/A001353(n+1).

Original entry on oeis.org

1, 14, 161, 1792, 19809, 218638, 2412353, 26614784, 293628097, 3239445006, 35739069409, 394290020096, 4349990523425, 47991114171406, 529460241815169, 5841251080892416, 64443392518654337, 710969410782059534

Offset: 0

R. J. Mathar, Jun 03 2009

Proposed by R. Guy in the seqfan list Mar 28 2009.

With an offset of 1, this sequence is the case P1 = 14, P2 = 32, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Apr 27 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..900
H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.
H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume.
Index entries for linear recurrences with constant coefficients, signature (14,-34,14,-1).

Cf. A001563, A003696, A100047.

a:=[1,14,161,1792];; for n in [5..20] do a[n]:=14*a[n-1]-34*a[n-2] +14*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Dec 24 2019

I:=[1,14,161,1792]; [n le 4 select I[n] else 14*Self(n-1)-34*Self(n-2) +14*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Apr 28 2014

seq(simplify( ChebyshevU(n, (4+sqrt(2))/2)*ChebyshevU(n, (4-sqrt(2))/2) ), n = 0 .. 20); # G. C. Greubel, Dec 24 2019

CoefficientList[Series[(1-x^2)/(1-14x+34x^2-14x^3+x^4), {x, 0, 20}], x] (* Vincenzo Librandi, Apr 28 2014 *)
Table[Simplify[ChebyshevU[n, (4+Sqrt[2])/2]*ChebyshevU[n, (4-Sqrt[2])/2]], {n, 0, 20}] (* G. C. Greubel, Dec 24 2019 *)

vector(21, n, round(polchebyshev(n-1, 2, (4+sqrt(2))/2)*polchebyshev(n-1, 2, (4-sqrt(2))/2)) ) \\ G. C. Greubel, Dec 24 2019

[round(chebyshev_U(n,(4+sqrt(2))/2)*chebyshev_U(n,(4-sqrt(2))/2)) for n in (0..20)] # G. C. Greubel, Dec 24 2019

a(n) = 14*a(n-1) -34*a(n-2) +14*a(n-3) -a(n-4).
G.f.: (1-x^2)/(1-14*x+34*x^2-14*x^3+x^4).
From Peter Bala, Apr 27 2014: (Start)
The following remarks assume an offset of 1.
a(n) = (1/sqrt(17))*( T(n,(7 + sqrt(17))/2) - T(n,(7 - sqrt(17))/2) ), where T(n,x) is the Chebyshev polynomial of the first kind.
a(n) = the bottom left entry of the 2 X 2 matrix T(n,M), where M is the 2 X 2 matrix [0, -8; 1, 7].
a(n) = U(n-1,1/2*(4 + sqrt(2)))*U(n-1,1/2*(4 - sqrt(2))), where U(n,x) is the Chebyshev polynomial of the second kind.
See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

A188914 a(n) = n*n! + 1 = (n+1)! - n! + 1.

Original entry on oeis.org

1, 2, 5, 19, 97, 601, 4321, 35281, 322561, 3265921, 36288001, 439084801, 5748019201, 80951270401, 1220496076801, 19615115520001, 334764638208001, 6046686277632001, 115242726703104001, 2311256907767808001, 48658040163532800001, 1072909785605898240001

Offset: 0

John M. Campbell, Apr 17 2011

It is unknown if all numbers of the form n*n!+1 are squarefree. n*n!+1 is squarefree for 0 < n < 52. It is unknown if there exist infinitely many primes of the form n*n!+1. For primes in this sequence, see A049984.

Cf. A001563, A049984, A094258.

Mathematica
```
Table[(n*Factorial[n])+1,{n,0,30}]
```

a(n) = n*n! + 1; \\ Michel Marcus, Aug 03 2022

E.g.f.: exp(x) + x/(1 - x)^2. - Stefano Spezia, Aug 03 2022

a(0)=1 prepended by Alois P. Heinz, Aug 03 2022

cp's OEIS Frontend

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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A061018 Triangle: a(n,m) = number of permutations of (1,2,...,n) with one or more fixed points in the m first positions.

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A069149 Numbers k such that k*k!/A062758(k) is an integer where A062758(k) is the product of squares of divisors of k.

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A094304 Sum of all possible sums formed from all but one of the previous terms, starting 1.

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A145572 Numerators of partial sums for Liouville's constant, read as base 2 (binary) numbers.

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A154306 a(n) = (n+1)^3*(3+n)!/6.

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A154307 a(n) = (n+1)^4*(4+n)!/24.

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