

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
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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 (18391914).  Amiram Eldar, Jun 11 2021


REFERENCES

Donald E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.5.


LINKS



FORMULA

a(n) = Sum_{k=0..n} binomial(n,k)*Bell(2*nk).
a(n) = Sum_{k=0..n} (1)^k*binomial(n, k)*Bell(2*nk+1).
a(n) = exp(1)*Sum_{k>=0} (k(k+1))^n/k!.  Benoit Cloitre, Dec 30 2005


EXAMPLE

n = 1, S = {1, 2, 3}. k = n+1 = 2. Thus a(1) = card { 132, 123, 123 } = 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(m1)] for m > 0.
local A, R, M, n, k, j;
M := m+m1; 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[k1] := A[k1] + A[k]
od;
if is(n, odd) then
j := j+1; R := R, A[j] fi
od;
[R] end:


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]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



