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A094577 Central Peirce numbers. Number of set partitions of {1,2,..,2n+1} in which n+1 is the smallest of its block. 13
1, 3, 27, 409, 9089, 272947, 10515147, 501178937, 28773452321, 1949230218691, 153281759047387, 13806215066685433, 1408621900803060705, 161278353358629226675, 20555596673435403499083, 2896227959507289559616217, 448371253145121338801335489 (list; graph; refs; listen; history; text; 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. Peirce (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
Named after the American philosopher, logician, mathematician and scientist Charles Sanders Peirce (1839-1914). - Amiram Eldar, Jun 11 2021
REFERENCES
Donald E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.5.
LINKS
Charles Sanders Peirce, On the Algebra of Logic, American Journal of Mathematics, Vol. 3 (1880), pp. 15-57.
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k)*Bell(2*n-k).
a(n) = 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, Dec 30 2005
a(n) = Sum_{k=0..n} binomial(n,k)*Bell(n+k). - Vaclav Kotesovec, Jul 29 2022
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, 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]
PROG
(Python)
# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A094577_list, blist, b = [1], [1], 1
for n in range(2, 502):
....blist = list(accumulate([b]+blist))
....b = blist[-1]
....blist = list(accumulate([b]+blist))
....b = blist[-1]
....A094577_list.append(blist[-n])
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014
CROSSREFS
Main diagonal of array in A011971.
Sequence in context: A365586 A201696 A011781 * A221624 A108525 A136719
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, May 12 2004
STATUS
approved

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Last modified July 19 01:31 EDT 2024. Contains 374388 sequences. (Running on oeis4.)