login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Triangle read by rows: T(i,j) for the recurrence T(i,j) = (T(i-1,j) + 1)*i.
4

%I #18 Jun 22 2022 17:39:04

%S 1,4,2,15,9,3,64,40,16,4,325,205,85,25,5,1956,1236,516,156,36,6,13699,

%T 8659,3619,1099,259,49,7,109600,69280,28960,8800,2080,400,64,8,986409,

%U 623529,260649,79209,18729,3609,585,81,9,9864100,6235300,2606500,792100,187300,36100,5860,820,100,10

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

%C 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)).

%C 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

%F From _Manfred Boergens_, Jun 22 2022: (Start)

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

%F 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.

%F (End)

%e Triangle T(i,j) begins:

%e 1

%e 4 2

%e 15 9 3

%e 64 40 16 4

%e 325 205 85 25 5

%e 1956 1236 516 156 36 6

%e 13699 8659 3619 1099 259 49 7

%e ...

%p T:= proc(i, j) option remember;

%p `if`(j<1 or j>i, 0, T(i-1, j)*i+i)

%p end:

%p seq(seq(T(n, k), k=1..n), n=1..10); # _Alois P. Heinz_, Jun 22 2022

%t Table[Sum[m!/(m - i)!, {i, n}], {m, 9}, {n, m, 1, -1}] // Flatten (* _Michael De Vlieger_, Apr 22 2017 *)

%t (* Sum-free code *)

%t b[j_] = If[j == 0, 0, Floor[j! E - 1]];

%t T[i_, j_] = b[i] - i! b[j - 1]/(j - 1)!;

%t Table[T[i, j], {i, 24}, {j, i}] // Flatten

%t (* _Manfred Boergens_, Jun 22 2022 *)

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

%Y Mirror of triangle A285268.

%K nonn,tabl

%O 1,2

%A _Thomas Wieder_, Aug 15 2006

%E Edited by _N. J. A. Sloane_, Sep 15 2006

%E Formula in name corrected by _Alois P. Heinz_, Jun 22 2022