The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A283424 Number T(n,k) of blocks of size >= k in all set partitions of [n], assuming that every set partition contains one block of size zero; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 2, 1, 5, 3, 1, 15, 10, 4, 1, 52, 37, 17, 5, 1, 203, 151, 76, 26, 6, 1, 877, 674, 362, 137, 37, 7, 1, 4140, 3263, 1842, 750, 225, 50, 8, 1, 21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1, 115975, 94828, 57568, 25996, 8944, 2392, 502, 82, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS T(n,k) is defined for all n,k >= 0.  The triangle contains only the terms with k<=n.  T(n,k) = 0 for k>n. LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition of a set FORMULA T(n,k) = Sum_{j=0..n-k} binomial(n,j) * Bell(j). T(n,k) = Bell(n+1) - Sum_{j=0..k-1} binomial(n,j) * Bell(n-j). T(n,k) = Sum_{j=k..n} A056857(n+1,j) = Sum_{j=k..n} A056860(n+1,n+1-j). Sum_{k=0..n} T(n,k) = n*Bell(n)+Bell(n+1) = A124427(n+1). Sum_{k=1..n} T(n,k) = n*Bell(n) = A070071(n). T(n,0)-T(n,1) = Bell(n). EXAMPLE T(3,2) = 4 because the number of blocks of size >= 2 in all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+1+1+1+0 = 4. Triangle T(n,k) begins:       1;       2,     1;       5,     3,    1;      15,    10,    4,    1;      52,    37,   17,    5,    1;     203,   151,   76,   26,    6,   1;     877,   674,  362,  137,   37,   7,  1;    4140,  3263, 1842,  750,  225,  50,  8, 1;   21147, 17007, 9991, 4307, 1395, 345, 65, 9, 1;   ... MAPLE T:= proc(n, k) option remember; `if`(k>n, 0,       binomial(n, k)*combinat[bell](n-k)+T(n, k+1))     end: seq(seq(T(n, k), k=0..n), n=0..14); MATHEMATICA T[n_, k_] := Sum[Binomial[n, j]*BellB[j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2018 *) CROSSREFS Columns k=0-10 give: A000110(n+1), A138378 or A005493(n-1), A124325, A288785, A288786, A288787, A288788, A288789, A288790, A288791, A288792. Row sums give A124427(n+1). T(2n,n) gives A286896. Cf. A005493, A056857, A056860, A070071, A285595. Sequence in context: A349934 A188416 A160185 * A188392 A143409 A197387 Adjacent sequences:  A283421 A283422 A283423 * A283425 A283426 A283427 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 14 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 7 10:19 EDT 2022. Contains 355985 sequences. (Running on oeis4.)