A180119 -id:A180119 - cp's OEIS frontend

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.

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).

A285793 Sum T(n,k) of the k-th entries in all 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, 18, 13, 5, 96, 83, 43, 18, 600, 582, 342, 192, 84, 4320, 4554, 2874, 1824, 1068, 480, 35280, 39672, 26232, 17832, 11784, 7080, 3240, 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200, 3265920, 4044240, 2834640, 2078640, 1534320, 1110960, 765360, 473760, 221760

Offset: 1

Alois P. Heinz, Apr 26 2017

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

			T(3,2) = 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.
Triangle T(n,k) begins:
:      1;
:      4,      2;
:     18,     13,      5;
:     96,     83,     43,     18;
:    600,    582,    342,    192,     84;
:   4320,   4554,   2874,   1824,   1068,   480;
:  35280,  39672,  26232,  17832,  11784,  7080,  3240;
: 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;

Wikipedia, Permutation

Columns k=1-2 give: A001563, A285795.
Main diagonal and first lower diagonal give: A038720(n-1) (for n>1), A286175.
Row sums give A000142 * A000217 = A180119.
Cf. A285439, A285595, A286231.

T(n,1) = n * n!.

T(n,n) = floor((n-1)!*(n+2)/2).

A284816 Sum of entries in the first cycles of all permutations of [n].

Original entry on oeis.org

1, 4, 21, 132, 960, 7920, 73080, 745920, 8346240, 101606400, 1337212800, 18920563200, 286442956800, 4620449433600, 79114299264000, 1433211107328000, 27387931963392000, 550604138692608000, 11617107089043456000, 256671161862635520000, 5926549291918295040000

Offset: 1

Alois P. Heinz, Apr 15 2017

nonn

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

Also, the number of colorings of n+1 given balls, two thereof identical, using n given colors (each color is used). - Ivaylo Kortezov, Jan 27 2024

			a(3) = 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.

Alois P. Heinz, Table of n, a(n) for n = 1..448
Ivaylo Kortezov, Winter Math Contest Yambol 2024, Bulgaria (in Bulgarian), Problem 8.3.
Wikipedia, Permutation.

Cf. A006595, A180119, A185105, A285363, A285382.

Column k=1 of A285439.

a:= n-> n!*(n*(n+1)-(n-1)*(n+2)/2)/2:
seq(a(n), n=1..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, n,
       (n^2+n+2)*n*a(n-1)/(n^2-n+2))
    end:
seq(a(n), n=1..25);

a(n) = n!*(n*(n+1) - (n-1)*(n+2)/2)/2.
E.g.f.: -x*(x^2-2*x+2)/(2*(x-1)^3).
a(n) = (n^2+n+2)*n*a(n-1)/(n^2-n+2) for n > 1, a(n) = n for n < 2.
a(n) = n*A006595(n-1). - Ivaylo Kortezov, Feb 02 2024

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.

A300559 a(n) = n*(n+1)!/2 + 1.

Original entry on oeis.org

1, 2, 7, 37, 241, 1801, 15121, 141121, 1451521, 16329601, 199584001, 2634508801, 37362124801, 566658892801, 9153720576001, 156920924160001, 2845499424768001, 54420176498688001, 1094805903679488001, 23112569077678080001, 510909421717094400001, 11802007641664880640001

Offset: 0

M. F. Hasler, Apr 10 2018

See A301373 and A302859 for the primes: it is remarkable that all of a(1..10) are primes, and only a(11) is the first composite term.

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Inspired by A302859.

Cf. A301373.

[n*Factorial(n+1)/2+1: n in [0..25]]; // Vincenzo Librandi, Apr 12 2018

Array[# (# + 1)!/2 + 1 &, 22, 0] (* Michael De Vlieger, Apr 10 2018 *)

apply( A300559=n->(n+1)!\2*n+1, [0..25])

a(n) = A180119(n) + 1 = A001286(n+1) + 1.
D-finite with recurrence n*a(n+1) = (n+1)*(n+2)*(a(n)-1) + n. - Chai Wah Wu, Apr 11 2018
E.g.f.: exp(x)-1/(x-1)^3*x. - Simon Plouffe, Jun 21 2018

A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 19, 24, 251, 374, 953, 1104, 1507, 3390, 4443, 5762

Offset: 1

Daniel Suteu, Apr 03 2018

The associated primes are A300559(a(n)) = A180119(a(n))+1 = A001286(a(n)+1)+1. - M. F. Hasler, Apr 10 2018
Looking for primes of the form p(n) = 1 + n! f(n) with a simple polynomial function f, it appears that the choice f(n) = n(n+1)/2 = A000217 is one of the most successful choices for getting a maximum of primes for n = 1..20. - M. F. Hasler, Apr 14 2018
The PFGW program has been used to certify all the terms up to a(23), using a deterministic test which exploits the factorization of a(n) - 1. - Giovanni Resta, Jun 24 2018

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Cf. A090703, A300559, A180119, A001286.

See A302859 for the actual primes.

Do[ If[ PrimeQ[n(n +1)!/2 +1], Print@ n], {n, 4000}] (* Robert G. Wilson v, Apr 05 2018 *)

isok(k) = ispseudoprime((k+1)! * k / 2 + 1);

a(21) from Robert G. Wilson v, Apr 05 2018
a(22) from Vaclav Kotesovec, Apr 06 2018
a(23) from Giovanni Resta, Jun 24 2018

A156815 Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 2, 6, 6, 0, 6, 28, 36, 24, 0, 24, 180, 300, 240, 120, 0, 120, 1488, 3240, 3120, 1800, 720, 0, 720, 15120, 43344, 50400, 33600, 15120, 5040, 0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320, 0, 40320, 2570400, 13068000, 22377600, 20018880, 11430720, 4656960, 1451520, 362880

Offset: 0

Roger L. Bagula, Feb 16 2009

			Triangle begins as:
  1;
  0,    1;
  0,    1,      2;
  0,    2,      6,      6;
  0,    6,     28,     36,     24;
  0,   24,    180,    300,    240,    120;
  0,  120,   1488,   3240,   3120,   1800,    720;
  0,  720,  15120,  43344,  50400,  33600,  15120,   5040;
  0, 5040, 182880, 695520, 979776, 756000, 383040, 141120, 40320;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), page 99.

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

Cf. A048993, A029767, A180119.

[Factorial(n)*StirlingSecond(n,k)/Binomial(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 10 2021

T[n_, k_] = n!*StirlingS2[n, k]/Binomial[n, k];
Table[T[n, k], {n, 0, 12}, {k,0,n}]//Flatten

flatten([[factorial(n)*stirling_number2(n,k)/binomial(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 10 2021

T(n, k) = n!*StirlingS2(n, k)/binomial(n, k).
From G. C. Greubel, Jun 10 2021: (Start)
T(n, 1) = T(n, n) = n!.
T(n, 2) = 2*A029767(n+1).
T(n, n-1) = A180119(n). (End)

Edited by G. C. Greubel, Jun 10 2021

A305739 a(n) = n!*T(n) - 1, where T(n) is the n-th triangular number.

Original entry on oeis.org

0, 5, 35, 239, 1799, 15119, 141119, 1451519, 16329599, 199583999, 2634508799, 37362124799, 566658892799, 9153720575999, 156920924159999, 2845499424767999, 54420176498687999, 1094805903679487999, 23112569077678079999, 510909421717094399999

Offset: 1

Maheswara Rao Valluri, Jun 22 2018

nonn

Cf. A000142, A000217, A001286, A180119.

See A305738 for the indices of primes in this sequence.

Maple
```
seq(n*(n+1)!/2-1,n=1..21);
```

a(n) = n*(n+1)!/2 - 1; \\ Michel Marcus, Jun 23 2018

a(n) = n*(n+1)!/2 - 1. - Michel Marcus, Jun 23 2018

a(n) = A180119(n)-1 = A001286(n+1)-1. - Alois P. Heinz, Jun 24 2018

cp's OEIS Frontend

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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A285793 Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A284816 Sum of entries in the first cycles of all permutations of [n].

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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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A300559 a(n) = n*(n+1)!/2 + 1.

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A301373 Numbers k such that (k+1)!*k/2 + 1 is prime.

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A156815 Triangle T(n, k) = n!*StirlingS2(n, k)/binomial(n, k), read by rows.

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A305739 a(n) = n!*T(n) - 1, where T(n) is the n-th triangular number.

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