A121662 Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.

1, 4, 2, 15, 9, 3, 64, 40, 16, 4, 325, 205, 85, 25, 5, 1956, 1236, 516, 156, 36, 6, 13699, 8659, 3619, 1099, 259, 49, 7, 109600, 69280, 28960, 8800, 2080, 400, 64, 8, 986409, 623529, 260649, 79209, 18729, 3609, 585, 81, 9, 9864100, 6235300, 2606500, 792100, 187300, 36100, 5860, 820, 100, 10

Offset

1,2

Comments

The first column is A007526 = "the number of (nonnull) "variations" of n distinct objects, namely the number of permutations of nonempty subsets of {1,...,n}." E.g. for n=3 there are 15 subsets: {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}, {abc}, {acb}, {bac}, {bca}, {cab}, {cba}. These are subsets with a number of elements l=1,...,n. The second column excludes all subsets with l=n elements. For n=3 one has therefore only the 9 subsets {a}, {b}, {c}, {ab}, {ba}, {ac}, {ca}, {bc}, {cb}. The third column excludes all subsets with l>=n-1 elements. For n=3 one has therefore only the 3 subsets {a}, {b},{c}. See also A121684. The second column is A038156 = n!*Sum(1/k!, k=1..n-1). The first lower diagonal are the squares A000290 = n^2. The second lower diagonal (15, 40, 85...) is A053698 = n^3 + n^2 + n + 1. The row sum is A030297 = a(n) = n*(n+a(n-1)).

T(i, j) is the total number of ordered sets of size 1 to i-j+1 that can be created from i distinct items. - Manfred Boergens, Jun 22 2022

Links

Table of n, a(n) for n=1..55.

Formula

From Manfred Boergens, Jun 22 2022: (Start)

T(i, j) = Sum_{k=1..i-j+1} i!/(i-k)! = Sum_{k=j-1..i-1} i!/k!.

Sum-free formula: T(i, j) = b(i) - i!*b(j-1)/(j-1)! where b(0)=0, b(j)=floor(j!*e-1) for j>0.

(End)

Example

Triangle T(i,j) begins:
       1
       4     2
      15     9     3
      64    40    16     4
     325   205    85    25    5
    1956  1236   516   156   36   6
   13699  8659  3619  1099  259  49  7
   ...

Maple

T:= proc(i, j) option remember;
      `if`(j<1 or j>i, 0, T(i-1, j)*i+i)
    end:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Jun 22 2022

Mathematica

Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m, 1, -1}] // Flatten (* Michael De Vlieger, Apr 22 2017 *)
(* Sum-free code *)
b[j_] = If[j == 0, 0, Floor[j! E - 1]];
T[i_, j_] = b[i] - i! b[j - 1]/(j - 1)!;
Table[T[i, j], {i, 24}, {j, i}] // Flatten
(* Manfred Boergens, Jun 22 2022 *)

Crossrefs

Cf. A007526, A030297, A053698, A038156, A000290, A121682.

Mirror of triangle A285268.

Sequence in context: A261870 A325516 A299789 * A130042 A285362 A109922

Adjacent sequences: A121659 A121660 A121661 * A121663 A121664 A121665

Keyword

nonn,tabl

Author

Thomas Wieder, Aug 15 2006

Extensions

Edited by N. J. A. Sloane, Sep 15 2006

Formula in name corrected by Alois P. Heinz, Jun 22 2022

Status

approved