A285793 -id:A285793 - cp's OEIS frontend

A038720 a(n) = (n+3)*n!/2.

2, 5, 18, 84, 480, 3240, 25200, 221760, 2177280, 23587200, 279417600, 3592512000, 49816166400, 741015475200, 11769069312000, 198766503936000, 3556874280960000, 67224923910144000, 1338096104497152000, 27978373094031360000, 613091306060513280000

Offset: 1

N. J. A. Sloane, May 02 2000

Next-to-last diagonal of A038719.

a(n-1) is the sum of the n-th entries in all cycles of all permutations of [n]. a(2) = 5 because the sum of the third entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 3+2+0+0+0+0 = 5. - Alois P. Heinz, May 03 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Rajesh Kumar Mohapatra and Tzung-Pei Hong, On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences, Mathematics (2022) Vol. 10, No. 7, 1161.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.
Index entries for sequences related to posets.

Cf. A038719, A038721, A052572, A214178.

Main diagonal of A285793.

import Data.List (transpose)
a038720 n = a038720_list !! (n-1)
a038720_list = (transpose $ map reverse a038719_tabl) !! 1
-- Reinhard Zumkeller, Jul 08 2012

A038720:= func< n | (n+3)*Factorial(n)/2 >; // G. C. Greubel, May 11 2025

Array[(# + 3) #!/2 &, 21] (* Michael De Vlieger, Apr 28 2022 *)

def A038720(n): return (n+3)*factorial(n)//2 # G. C. Greubel, May 11 2025

a(n) = A052572(n)/2.
a(n) = A214178(n+3,n). - Reinhard Zumkeller, Jul 08 2012
G.f.: Sum_{n>=1} ( (n+1)*x/(1 + (n+1)*x) )^n. - Paul D. Hanna, Jan 02 2013
E.g.f.: 1/(1-x) + 1/(2*(x-1)^2) - 3/2. - Alois P. Heinz, May 04 2017
From Amiram Eldar, Dec 11 2022: (Start)
Sum_{n>=1} 1/a(n) = 2*e - 14/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 10/e - 10/3. (End)

Corrected and extended by Larry Reeves (larryr(AT)acm.org), May 09 2000.

A180119 a(n) = (n+2)! * Sum_{k = 1..n} 1/((k+1)*(k+2)).

Original entry on oeis.org

0, 1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000, 510909421717094400000, 11802007641664880640000

Offset: 0

Gary Detlefs, Aug 10 2010

In general, a sequence of the form (n+x+2)! * Sum_{k = 1..n} (k+x)!/(k+x+2)! will have a closed form of (n+x+2)!*n/((x+2)*(n+2+x)).
0 followed by A001286. - R. J. Mathar, Aug 13 2010
Sum of the entries in all cycles of all permutations of [n]. - Alois P. Heinz, Apr 19 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..300
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 209. Book's website
H. W. Gould, ed. J. Quaintance, Combinatorial Identities, May 2010 (section 10, p.45).

Cf. A000142, A000217, A001286, A037960, A037961, A285439, A285793.

[n*Factorial(n+2)/(2*(n+2)): n in [0..25]]; // Vincenzo Librandi, Feb 20 2017

a:= n-> n*(n+2)!/(2*(n+2)): seq(a(n), n=0..20);

Table[n (n + 2)! / (2 (n + 2)), {n, 0, 30}] (* Vincenzo Librandi, Feb 20 2017 *)

a(n) = (n+2)! * sum(k=1, n,1/((k+1)*(k+2))); \\ Michel Marcus, Jan 10 2015

apply( A180119(n)=(n+1)!\2*n, [0..20]) \\ M. F. Hasler, Apr 10 2018

a(n) = n*(n+1)!/2. [Simplified by M. F. Hasler, Apr 10 2018]
a(n) = (n+1)! * Sum_{k = 2..n} (1/(k^2+k)), with offset 1. - Gary Detlefs, Sep 15 2011
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*k^(n+1) = (1/(2*x + 1))*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(x*n + k)^(n+1), for arbitrary x != -1/2. - Peter Bala, Feb 19 2017
From Alois P. Heinz, Apr 19 2017: (Start)
a(n) = A000142(n) * A000217(n) = Sum_{k=1..n} A285439(n,k).
E.g.f.: x/(1-x)^3. (End)
a(n) = A001286(n+1) for n > 0. - M. F. Hasler, Apr 10 2018

A285439 Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 21, 12, 3, 132, 76, 28, 4, 960, 545, 235, 55, 5, 7920, 4422, 2064, 612, 96, 6, 73080, 40194, 19607, 6692, 1386, 154, 7, 745920, 405072, 202792, 75944, 18736, 2816, 232, 8, 8346240, 4484808, 2280834, 911637, 254061, 46422, 5256, 333, 9

Offset: 1

Alois P. Heinz, Apr 19 2017

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

			T(3,1) = 21 because the sum of the entries in the first cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 6+6+3+4+1+1 = 21.
Triangle T(n,k) begins:
       1;
       4,      2;
      21,     12,      3;
     132,     76,     28,     4;
     960,    545,    235,    55,     5;
    7920,   4422,   2064,   612,    96,    6;
   73080,  40194,  19607,  6692,  1386,  154,   7;
  745920, 405072, 202792, 75944, 18736, 2816, 232, 8;
  ...

Alois P. Heinz, Rows n = 1..23, flattened
Wikipedia, Permutation

Columns k=1-2 give: A284816, A285489.
Row sums give A000142 * A000217 = A180119.
Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
Cf. A000290, A002775, A185105, A285362, A285382, A285793.

T:= proc(h) option remember; local b; b:=
      proc(n, l) option remember; `if`(n=0, [mul((i-1)!, i=l), 0],
        (p-> p+[0, (h-n+1)*p[1]*x^(nops(l)+1)])(b(n-1, [l[], 1]))+
         add((p-> p+[0, (h-n+1)*p[1]*x^j])(
         b(n-1, subsop(j=l[j]+1, l))), j=1..nops(l)))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
    end:
seq(T(n), n=1..10);

T[h_] := T[h] = Module[{b}, b[n_, l_] := b[n, l] = If[n == 0, {Product[(i - 1)!, {i, l}], 0}, # + {0, (h - n + 1)*#[[1]]*x^(Length[l] + 1)}&[b[n - 1, Append[l, 1]]] + Sum[# + {0, (h-n+1)*#[[1]]*x^j}&[b[n - 1, ReplacePart[ l, j -> l[[j]] + 1]]], {j, 1, Length[l]}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, {}][[2]]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 25 2018, translated from Maple *)

Sum_{k=1..n} k * T(n,k) = n^2 * n! = A002775(n).

A285595 Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 4, 2, 17, 10, 3, 76, 52, 18, 4, 362, 274, 111, 28, 5, 1842, 1500, 675, 200, 40, 6, 9991, 8614, 4185, 1380, 325, 54, 7, 57568, 51992, 26832, 9568, 2510, 492, 70, 8, 351125, 329650, 178755, 67820, 19255, 4206, 707, 88, 9, 2259302, 2192434, 1239351, 494828, 149605, 35382, 6629, 976, 108, 10

Offset: 1

Alois P. Heinz, Apr 22 2017

T(n,k) is also k times the number of blocks of size >k in all set partitions of [n+1]. T(3,2) = 10 = 2 * 5 because there are 5 blocks of size >2 in all set partitions of [4], namely in 1234, 123|4, 124|3, 134|2, 1|234.

			T(3,2) = 10 because the sum of the second entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+2+3+3+0  = 10.
Triangle T(n,k) begins:
      1;
      4,     2;
     17,    10,     3;
     76,    52,    18,    4;
    362,   274,   111,   28,    5;
   1842,  1500,   675,  200,   40,   6;
   9991,  8614,  4185, 1380,  325,  54,  7;
  57568, 51992, 26832, 9568, 2510, 492, 70, 8;
  ...

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

Column k=1 gives A124325(n+1).
Row sums give A000110(n) * A000217(n) = A105488(n+3).
Main diagonal and first lower diagonal give: A000027, A028552.
Cf. A007318, A175757, A278677, A283424, A285362, A285793, A286897.

T:= proc(h) option remember; local b; b:=
      proc(n, l) option remember; `if`(n=0, [1, 0],
        (p-> p+[0, (h-n+1)*p[1]*x^1])(b(n-1, [l[], 1]))+
         add((p-> p+[0, (h-n+1)*p[1]*x^(l[j]+1)])(b(n-1,
         sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
      end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
    end:
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*add(x^k, k=1..j-1)])(
         b(n-j)*binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i)*i, i=1..n))(b(n+1)[2]):
seq(T(n), n=1..12);

b[n_] := b[n] = If[n == 0, {1, 0}, Sum[# + {0, #[[1]]*Sum[x^k, {k, 1, j-1} ]}&[b[n - j]*Binomial[n - 1, j - 1]], {j, 1, n}]];
T[n_] := Table[Coefficient[#, x, i]*i, {i, 1, n}] &[b[n + 1][[2]]];
Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, May 23 2018, translated from 2nd Maple program *)

T(n,k) = k * Sum_{j=k+1..n+1} binomial(n+1,j)*A000110(n+1-j).
T(n,k) = k * Sum_{j=k+1..n+1} A175757(n+1,j).
Sum_{k=1..n} T(n,k)/k = A278677(n-1).

A286231 Sum T(n,k) of the entries in the k-th last cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 5, 1, 25, 10, 1, 143, 79, 17, 1, 942, 634, 197, 26, 1, 7074, 5462, 2129, 417, 37, 1, 59832, 51214, 23381, 5856, 786, 50, 1, 563688, 523386, 269033, 80053, 13934, 1360, 65, 1, 5858640, 5813892, 3281206, 1111498, 232349, 29728, 2204, 82, 1

Offset: 1

Alois P. Heinz, May 04 2017

			T(3,2) = 10 because the sum of the entries in the second last cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 0+0+3+4+1+2 = 10.
Triangle T(n,k) begins:
       1;
       5,      1;
      25,     10,      1;
     143,     79,     17,     1;
     942,    634,    197,    26,     1;
    7074,   5462,   2129,   417,    37,    1;
   59832,  51214,  23381,  5856,   786,   50,  1;
  563688, 523386, 269033, 80053, 13934, 1360, 65, 1;
  ...

Alois P. Heinz, Rows n = 1..20, flattened
Wikipedia, Permutation

Column k=1 gives A285382.
Main diagonal and first lower diagonal give: A000012, A002522.
Row sums give A000142 * A000217 = A180119.
Cf. A285793, A286232.

A285795 Sum of the second entries in all cycles of all permutations of [n].

Original entry on oeis.org

0, 0, 2, 13, 83, 582, 4554, 39672, 382248, 4044240, 46663920, 583554240, 7865622720, 113711230080, 1755484617600, 28828769356800, 501858148377600, 9232213174732800, 178968924600883200, 3646603415927808000, 77916767838981120000, 1742147265551616000000

Offset: 0

Alois P. Heinz, Apr 26 2017

nonn

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

			a(3) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.

Alois P. Heinz, Table of n, a(n) for n = 0..449
Wikipedia, Permutation

Column k=2 of A285793.

Cf. A001008, A002805.

a:= proc(n) option remember; `if`(n<3, n*(n-1),
     ((2*n^2+3*n-1)*a(n-1)-(n-1)*n*(n+2)*a(n-2))/(n+1))
    end:
seq(a(n), n=0..25);

Flatten[{0, Table[n! * (2*(n+1)*HarmonicNumber[n] - n - 3)/4, {n, 1, 25}]}] (* Vaclav Kotesovec, Apr 29 2017 *)

A286175 Sum of the n-th entries in all cycles of all permutations of [n+1].

Original entry on oeis.org

4, 13, 43, 192, 1068, 7080, 54360, 473760, 4616640, 49714560, 586051200, 7504358400, 103703846400, 1538074137600, 24366332390400, 410609751552000, 7333437855744000, 138362409529344000, 2749819506610176000, 57416487392968704000, 1256593887223234560000

Offset: 1

Alois P. Heinz, May 03 2017

nonn

			a(2) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.

Alois P. Heinz, Table of n, a(n) for n = 1..448

Cf. A285793.

a:= proc(n) option remember; `if`(n<3, [4, 13][n],
      (n-1)*(2*n^2+7*n+4)*a(n-1)/(2*n^2+3*n-1))
    end:
seq(a(n), n=1..25);

a[n_] := a[n] = If[n < 3, {4, 13}[[n]],
     (n-1)*(2*n^2 + 7*n + 4)*a[n-1]/(2*n^2 + 3*n - 1)];
Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Apr 21 2022, after Alois P. Heinz *)

E.g.f.: -2*log(1-x)-(5*x^3-10*x^2+10*x-7)/(2*(1-x)^2)-7/2.

a(n) = A285793(n+1,n).

cp's OEIS Frontend

A038720 a(n) = (n+3)*n!/2.

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A180119 a(n) = (n+2)! * Sum_{k = 1..n} 1/((k+1)*(k+2)).

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A285439 Sum T(n,k) of the entries in the k-th cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A285595 Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A286231 Sum T(n,k) of the entries in the k-th last cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A285795 Sum of the second entries in all cycles of all permutations of [n].

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A286175 Sum of the n-th entries in all cycles of all permutations of [n+1].

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