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A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements. 2
1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007

LINKS

Table of n, a(n) for n=1..66.

FORMULA

T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).

Double e.g.f.: exp(exp(x)*(exp(y)-1)).

U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007

EXAMPLE

Triangle T(n,k) starts:

1;

0,1;

0,1,2;

0,1,3,5;

0,1,5,10,15;

T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.

The arrays U(n,k) starts:

1 0 0 0 0 ...

1 1 1 1 1 ...

2 3 5 9 17 ...

5 10 22 52 130 ...

15 37 99 283 855 ...

CROSSREFS

Row sums of T(n, k) yield A124496(n, 1).

Cf. A108461. Columns of U(n, k): A000110, A005493, A033452. Rows of U(n, k): A000007, A000012, A000051. Main diagonal: A108459.

Sequence in context: A091612 A253672 A213861 * A254281 A295682 A195772

Adjacent sequences:  A108455 A108456 A108457 * A108459 A108460 A108461

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Jun 03 2005; Emeric Deutsch, Nov 14 2006

EXTENSIONS

Edited by N. J. A. Sloane, May 22 2008, at the suggestion of Vladeta Jovovic. This entry is a composite of two entries submitted independently by Christian G. Bower and Emeric Deutsch, with additional comments from Augustine O. Munagi.

STATUS

approved

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Last modified February 19 18:21 EST 2019. Contains 320327 sequences. (Running on oeis4.)