A322383 -id:A322383 - cp's OEIS frontend

A001563 a(n) = n*n! = (n+1)! - n!.

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

Offset: 0

N. J. A. Sloane

A similar sequence, with the initial 0 replaced by 1, namely A094258, is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002
Denominators in power series expansion of E_1(x) + gamma + log(x), x > 0. - Michael Somos, Dec 11 2002
If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g., there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3), ... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3), which rotates the last 1 element, i.e., it makes no change. Permutation 1 is (0,1,3,2), which rotates the last 2 elements. Permutation 4 is (0,3,1,2), which rotates the last 3 elements. Permutation 18 is (3,0,1,2), which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003
Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos, Mar 04 2004
From Michael Somos, Apr 27 2012: (Start)
Stirling transform of a(n)=[1,4,18,96,...] is A069321(n)=[1,5,31,233,...].
Partial sums of a(n)=[0,1,4,18,...] is A033312(n+1)=[0,1,5,23,...].
Binomial transform of A000166(n+1)=[0,1,2,9,...] is a(n)=[0,1,4,18,...].
Binomial transform of A000255(n+1)=[1,3,11,53,...] is a(n+1)=[1,4,18,96,...].
Binomial transform of a(n)=[0,1,4,18,...] is A093964(n)=[0,1,6,33,...].
Partial sums of A001564(n)=[1,3,4,14,...] is a(n+1)=[1,4,18,96,...].
(End)
Number of small descents in all permutations of [n+1]. A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1. Example: a(2)=4 because there are 4 small descents in the permutations 123, 13\2, 2\13, 231, 312, 3\2\1 of {1,2,3} (shown by \). a(n)=Sum_{k=0..n-1}k*A123513(n,k). - Emeric Deutsch, Oct 02 2006
Equivalently, in the notation of David, Kendall and Barton, p. 263, this is the total number of consecutive ascending pairs in all permutations on n+1 letters (cf. A010027). - N. J. A. Sloane, Apr 12 2014
a(n-1) is the number of permutations of n in which n is not fixed; equivalently, the number of permutations of the positive integers in which n is the largest element that is not fixed. - Franklin T. Adams-Watters, Nov 29 2006
Number of factors in a determinant when writing down all multiplication permutations. - Mats Granvik, Sep 12 2008
a(n) is also the sum of the positions of the left-to-right maxima in all permutations of [n]. Example: a(3)=18 because the positions of the left-to-right maxima in the permutations 123,132,213,231,312 and 321 of [3] are 123, 12, 13, 12, 1 and 1, respectively and 1+2+3+1+2+1+3+1+2+1+1=18. - Emeric Deutsch, Sep 21 2008
Equals eigensequence of triangle A002024 ("n appears n times"). - Gary W. Adamson, Dec 29 2008
Preface the series with another 1: (1, 1, 4, 18, ...); then the next term = dot product of the latter with "n occurs n times". Example: 96 = (1, 1, 4, 8) dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). - Gary W. Adamson, Apr 17 2009
Row lengths of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012
a(n) is also the number of minimum (n-)distinguishing labelings of the star graph S_{n+1} on n+1 nodes. - Eric W. Weisstein, Oct 14 2014
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right, i.e., a(n) is the permutation with the cycle notation (0 1 ... n-1 n). Compare array A051683 for circular shifts to the right in a broader sense. Compare sequence A007489 for circular shifts to the left. - Tilman Piesk, Apr 29 2017
a(n-1) is the number of permutations on n elements with no cycles of length n. - Dennis P. Walsh, Oct 02 2017
The number of pandigital numbers in base n+1, such that each digit appears exactly once. For example, there are a(9) = 9*9! = 3265920 pandigital numbers in base 10 (A050278). - Amiram Eldar, Apr 13 2020

			E_1(x) + gamma + log(x) = x/1 - x^2/4 + x^3/18 - x^4/96 + ..., x > 0. - _Michael Somos_, Dec 11 2002
G.f. = x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 37, equation 37:6:1 at page 354.

T. D. Noe, Table of n, a(n) for n = 0..100
J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 30.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
L. B. W. Jolley, Summation of Series, Dover, 1961.
I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008.
C. Lanczos, Applied Analysis (Annotated scans of selected pages).
Rezsö L. Lovas and István Mezö, On an exotic topology of the integers, arXiv:1008.0713 [math.GN], 2010. See p. 4.
Daniel J. Mundfrom, A problem in permutations: the game of `Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Dennis P. Walsh, The number of permutations with no k-cycles.
Eric Weisstein's World of Mathematics, Distinguishing Number.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A000142, A002024, A047920, A047922, A053495, A055089, A123513, A143946, A229837, A239069.
Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).
Cf. sequences with formula (n + k)*n! listed in A282466.
Row sums of A185105, A322383, A322384, A094485.

List([0..20], n-> n*Factorial(n) ); # G. C. Greubel, Dec 30 2019

a001563 n = a001563_list !! n
a001563_list = zipWith (-) (tail a000142_list) a000142_list
-- Reinhard Zumkeller, Aug 05 2013

[Factorial(n+1)-Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
A001563 := n->n*n!;
```

Table[n!n,{n,0,25}] (* Harvey P. Dale, Oct 03 2011 *)

{a(n) = if( n<0, 0, n * n!)} /* Michael Somos, Dec 11 2002 */

[n*factorial(n) for n in (0..20)] # G. C. Greubel, Dec 30 2019

From Michael Somos, Dec 11 2002: (Start)
E.g.f.: x / (1 - x)^2.
a(n) = -A021009(n, 1), n >= 0. (End)
The coefficient of y^(n-1) in expansion of (y+n!)^n, n >= 1, gives the sequence 1, 4, 18, 96, 600, 4320, 35280, ... - Artur Jasinski, Oct 22 2007
Integral representation as n-th moment of a function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(x*(x-1)*exp(-x)) dx, for n>=0. This representation may not be unique. - Karol A. Penson, Sep 27 2001
a(0)=0, a(n) = n*a(n-1) + n!. - Benoit Cloitre, Feb 16 2003
a(0) = 0, a(n) = (n - 1) * (1 + Sum_{i=1..n-1} a(i)) for i > 0. - Gerald McGarvey, Jun 11 2004
Arises in the denominators of the following identities: Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)) = 1/18, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)*(n+4)) = 1/96, etc. The general expression is Sum_{n>=k} 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005 [And the general expression implies that Sum_{n>=1} 1/(n*(n+1)*...*(n+k-1)) = (Sum_{n>=k} 1/C(n, k))/k! = 1/((k-1)*(k-1)!) = 1/a(k-1), k >= 2. - Jianing Song, May 07 2023]
a(n) = Sum_{m=2..n+1} |Stirling1(n+1, m)|, n >= 1 and a(0):=0, where Stirling1(n, m) = A048994(n, m), n >= m = 0.
a(n) = 1/(Sum_{k>=0} k!/(n+k+1)!), n > 0. - Vladeta Jovovic, Sep 13 2006
a(n) = Sum_{k=1..n(n+1)/2} k*A143946(n,k). - Emeric Deutsch, Sep 21 2008
The reciprocals of a(n) are the lead coefficients in the factored form of the polynomials obtained by summing the binomial coefficients with a fixed lower term up to n as the upper term, divided by the term index, for n >= 1: Sum_{k = i..n} C(k, i)/k = (1/a(n))*n*(n-1)*..*(n-i+1). The first few such polynomials are Sum_{k = 1..n} C(k, 1)/k = (1/1)*n, Sum_{k = 2..n} C(k, 2)/k = (1/4)*n*(n-1), Sum_{k = 3..n} C(k, 3)/k = (1/18)*n*(n-1)*(n-2), Sum_{k = 4..n} C(k, 4)/k = (1/96)*n*(n-1)*(n-2)*(n-3), etc. - Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) then a(n) = (-1)^(n-1)*f(n,1,-2), (n >= 1). - Milan Janjic, Mar 01 2009
Sum_{n>=1} (-1)^(n+1)/a(n) =  0.796599599... [Jolley eq. 289]
G.f.: 2*x*Q(0), where Q(k) = 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: W(0)*(1-sqrt(x)) - 1, where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 18 2013
G.f.: T(0)/x - 1/x, 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, Oct 17 2013
G.f.: Q(0)*(1-x)/x - 1/x, where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
D-finite with recurrence: a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, Jan 14 2020
a(n) = (-1)^(n+1)*(n+1)*Sum_{k=1..n} A094485(n,k)*Bernoulli(k). The inverse of the Worpitzky representation of the Bernoulli numbers. - Peter Luschny, May 28 2020
From Amiram Eldar, Aug 04 2020: (Start)
Sum_{n>=1} 1/a(n) = Ei(1) - gamma = A229837.
Sum_{n>=1} (-1)^(n+1)/a(n) = gamma - Ei(-1) = A239069. (End)
a(n) = Gamma(n)*A000290(n) for n > 0. - Jacob Szlachetka, Jan 01 2022

A322384 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 13, 4, 1, 67, 21, 7, 1, 411, 131, 46, 11, 1, 2911, 950, 341, 101, 16, 1, 23563, 7694, 2871, 932, 197, 22, 1, 213543, 70343, 26797, 9185, 2311, 351, 29, 1, 2149927, 709015, 275353, 98317, 27568, 5119, 583, 37, 1, 23759791, 7867174, 3090544, 1141614, 343909, 73639, 10366, 916, 46, 1

Offset: 1

Alois P. Heinz, Dec 05 2018

			The 6 permutations of {1,2,3} are:
  (1)     (2) (3)
  (1,2)   (3)
  (1,3)   (2)
  (2,3)   (1)
  (1,2,3)
  (1,3,2)
so there are 13 elements in the first cycles, 4 in the second cycles and only 1 in the third cycles.
Triangle T(n,k) begins:
       1;
       3,     1;
      13,     4,     1;
      67,    21,     7,    1;
     411,   131,    46,   11,    1;
    2911,   950,   341,  101,   16,   1;
   23563,  7694,  2871,  932,  197,  22,  1;
  213543, 70343, 26797, 9185, 2311, 351, 29, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Columns k=1-10 give: A028418, A332851, A332852, A332853, A332854, A332855, A332856, A332857, A332858, A332859.
Row sums give A001563.
T(2n,n) gives A332928.
Cf. A185105, A322383.

b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]]*x^i, {i, 1, Length[l]}], Sum[Binomial[n-1, j-1]*b[n-j, Sort[Append[l, j]]]*(j-1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, {}], x] // Rest;
Array[T, 12] // Flatten  (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 7, 7, 1, 21, 25, 13, 1, 66, 101, 71, 21, 1, 258, 366, 396, 166, 31, 1, 1079, 1555, 1877, 1247, 337, 43, 1, 4987, 7099, 9199, 7855, 3305, 617, 57, 1, 25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1, 136723, 184033, 253108, 284968, 203278, 79756, 16126, 1666, 91, 1

Offset: 1

Alois P. Heinz, Dec 07 2018

			The 5 set partitions of {1,2,3} are:
  1   |2  |3
  1   |23
  2   |13
  3   |12
  123
so there are 7 elements in the first (smallest) blocks, 7 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
      1;
      3,     1;
      7,     7,     1;
     21,    25,    13,     1;
     66,   101,    71,    21,     1;
    258,   366,   396,   166,    31,    1;
   1079,  1555,  1877,  1247,   337,   43,    1;
   4987,  7099,  9199,  7855,  3305,  617,   57,  1;
  25195, 34627, 47371, 47245, 27085, 7681, 1045, 73, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Wikipedia, Partition of a set

Column k=1-10 gives A097147, A332942, A332943, A332944, A332945, A332946, A332947, A332948, A332949, A332950.
Row sums give A070071.
Cf. A319375, A322383.

b:= proc(n, l) option remember; `if`(n=0, add(l[i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j])), j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
      max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
    end:
T:= (n, k)-> b(n, 1, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 02 2020

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)

A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

Original entry on oeis.org

1, 3, 10, 45, 236, 1505, 10914, 90601, 837304, 8610129, 96625970, 1184891081, 15665288484, 223149696601, 3394965018886, 55123430466945, 948479737691504, 17289345305870561, 332019600921360594, 6713316975465246889, 142321908843254560540, 3161718732648662557161

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..450

Cf. A028418, A046746, A006128.
Cf. A005225.
Column k=1 of A322383.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
b[n_, m_] := b[n, m] = If[n == 0, m, Sum[(j-1)! b[n-j, Min[m, j]]* Binomial[n-1, j-1], {j, n}]];
a[n_] := b[n, Infinity];
Array[a, 25] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

E.g.f.: Sum[k>0, -1+ exp(Sum(j>=k, x^j/j))]. - Vladeta Jovovic, Jul 26 2004

a(n) = Sum_{k=1..n} k * A145877(n,k). - Alois P. Heinz, Jul 28 2014

More terms from Vladeta Jovovic, Sep 19 2002

A332906 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 7, 37, 241, 1661, 13301, 117209, 1150297, 12314329, 144593989, 1828734689, 24995387561, 365311053953, 5707795873261, 94637770625761, 1665132643843201, 30896642665904609, 604541044692565157, 12416248460455779089, 267500866283111679289, 6024053249628809274769

Offset: 2

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=2 of A322383.

Cf. A349980.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 2)[2]:
seq(a(n), n=2..22);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i>n, 0, Sum[Function[ p, p + If[p =!= 0 && t>0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i-1)!^j* b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 2][[2]];
a /@ Range[2, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

a(n) = Sum_{k>=0} k * A349980(n,k). - Alois P. Heinz, Dec 07 2021

A332907 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 13, 101, 896, 7967, 78205, 827521, 9507454, 117211469, 1560454523, 22172178965, 336532052884, 5423997488041, 92726171603161, 1673203210233137, 31845893246619770, 636647098018469141, 13356074486442181999, 293166974869955073469, 6724854183662407594768

Offset: 3

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=3 of A322383.

Cf. A350016.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 3)[2]:
seq(a(n), n=3..22);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, 0, Sum[Function[ p, p + If[p =!= 0 && t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i - 1)!^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 3][[2]];
a /@ Range[3, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

a(n) = Sum_{k=0..n-2} k * A350016(n,k). - Alois P. Heinz, Dec 12 2021

A332908 Number of entries in the fourth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 21, 226, 2612, 29261, 346453, 4338214, 57819554, 815225643, 12234293579, 194294281572, 3264124624256, 57826690252441, 1079032037759257, 21142347350725466, 434563256137908638, 9344589765620199919, 209952915324112384719, 4919186923210370523448

Offset: 4

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=4 of A322383.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 4)[2]:
seq(a(n), n=4..22);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, 0, Sum[Function[ p, p + If[p =!= 0 && t>0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i-1)!^j* b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Array[i&, j], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 4][[2]];
a /@ Range[4, 22] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

A332909 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 31, 442, 6441, 88909, 1253104, 18332744, 280902678, 4497959259, 75694569341, 1336697348846, 24765423361061, 480653174845257, 9764210398405166, 207238383834819974, 4591419670284107644, 106002478632623159679, 2547169063966472089803, 63617191700084723716234

Offset: 5

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=5 of A322383.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 5)[2]:
seq(a(n), n=5..23);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][(i - 1)!^j*b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
a[n_] := b[n, 1, 5][[2]];
Table[a[n], {n, 5, 23}] (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)

A332910 Number of entries in the sixth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 43, 785, 14065, 234646, 3890624, 65790726, 1149152226, 20755593249, 389723127653, 7616467493381, 155034653456005, 3285524611730078, 72467273447829602, 1661985062849397148, 39610017536621076924, 979981089604037392299, 25145506663601266049403

Offset: 6

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=6 of A322383.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 6)[2]:
seq(a(n), n=6..23);

A332911 Number of entries in the seventh cycles of all permutations of [n] when cycles are ordered by increasing lengths.

Original entry on oeis.org

1, 57, 1297, 28006, 555446, 10718940, 207593322, 4101787080, 83030091003, 1731086543933, 37278557659029, 830606719626674, 19158608403920978, 457571632376980948, 11312327413794558348, 289427158238453113152, 7659193100381954895675, 209516361790095291013179

Offset: 7

Alois P. Heinz, Mar 02 2020

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..450
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Column k=7 of A322383.

b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
      add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))((i-1)!^j*
        b(n-i*j, i+1, max(0, t-j))/j!*combinat[multinomial]
         (n, i$j, n-i*j)), j=0..n/i)))
    end:
a:= n-> b(n, 1, 7)[2]:
seq(a(n), n=7..24);

cp's OEIS Frontend

A001563 a(n) = n*n! = (n+1)! - n!.

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A322384 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A319298 Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by increasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

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A332906 Number of entries in the second cycles of all permutations of [n] when cycles are ordered by increasing lengths.

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A332907 Number of entries in the third cycles of all permutations of [n] when cycles are ordered by increasing lengths.

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A332908 Number of entries in the fourth cycles of all permutations of [n] when cycles are ordered by increasing lengths.

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A332909 Number of entries in the fifth cycles of all permutations of [n] when cycles are ordered by increasing lengths.