A094577 Central Peirce numbers. Number of set partitions of {1,2,..,2n+1} in which n+1 is the smallest of its block.

1, 3, 27, 409, 9089, 272947, 10515147, 501178937, 28773452321, 1949230218691, 153281759047387, 13806215066685433, 1408621900803060705, 161278353358629226675, 20555596673435403499083, 2896227959507289559616217, 448371253145121338801335489

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

Alois P. Heinz, Table of n, a(n) for n = 0..288

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

Cf. A094574, A020556, A216078.

Main diagonal of array in A011971.

Sequence in context: A011781 A377717 A377693 * A221624 A108525 A136719

Adjacent sequences: A094574 A094575 A094576 * A094578 A094579 A094580

Keyword

nonn

Author

Vladeta Jovovic, May 12 2004

Status

approved