|
| |
|
|
A094577
|
|
Central Pierce numbers. Number of set partitions of {1,2,..,2n+1} in which n+1 is the smallest of its block.
|
|
5
| |
|
|
1, 3, 27, 409, 9089, 272947, 10515147, 501178937, 28773452321, 1949230218691, 153281759047387, 13806215066685433, 1408621900803060705, 161278353358629226675, 20555596673435403499083
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Let P(n,k) be the number of set partitions of {1,2,..,n} in which k is the smallest of its block. These numbers were introduced by C. S. Pierce (see reference, page 48). If this triangle is displayed as in A123346 (or A011971) then a(n) = A011971(2n, n) are the central Pierce numbers. - Peter Luschny, Jan 18 2011
|
|
|
REFERENCES
| D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.5.
C. S. Peirce, On the Algebra of Logic, American Journal of
Mathematics 3, (1880), 15 - 57.
|
|
|
FORMULA
| Sum_{k=0..n} binomial(n,k)*Bell(2*n-k). - Vladeta Jovovic
Or, Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k+1).
a(n)=exp(-1)*sum_{k>=0}(k(k+1))^n/k! - Benoit Cloitre (benoit7848c(AT)orange.fr), Dec 30 2005
|
|
|
EXAMPLE
| n = 1, S = {1, 2, 3}. k = n+1 = 2. Thus a(1) = card { 13|2, 1|23, 1|2|3 } = 3. - Peter Luschny, Jan 18 2011
|
|
|
MAPLE
| seq(add(binomial(n, k)*(bell(n+k)), k=0..n), n=0..14); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006
# The objective of this implementation is efficiency.
# m -> [a(0), a(1), ..., a(m-1)] for m > 0.
A094577_list := proc(m)
local A, R, M, n, k, j;
M := m+m-1; A := array(1..M);
j := 1; R := 1; A[1] := 1;
for n from 2 to M do
A[n] := A[1];
for k from n by -1 to 2 do
A[k-1] := A[k-1] + A[k]
od;
if is(n, odd) then
j := j+1; R := R, A[j] fi
od;
[R] end:
A094577_list(100); # example call - Peter Luschny, Jan 17 2011
|
|
|
MATHEMATICA
| f[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}]; Array[f, 15, 0]
|
|
|
CROSSREFS
| Cf. A094574, A020556.
Main diagonal of array in A011971.
Sequence in context: A141057 A201696 A011781 * A108525 A136719 A159600
Adjacent sequences: A094574 A094575 A094576 * A094578 A094579 A094580
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), May 12 2004
|
| |
|
|
