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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.

%I A094577 #51 May 22 2025 10:21:35
%S A094577 1,3,27,409,9089,272947,10515147,501178937,28773452321,1949230218691,
%T A094577 153281759047387,13806215066685433,1408621900803060705,
%U A094577 161278353358629226675,20555596673435403499083,2896227959507289559616217,448371253145121338801335489
%N A094577 Central Peirce numbers. Number of set partitions of {1,2,..,2n+1} in which n+1 is the smallest of its block.
%C A094577 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
%C A094577 Named after the American philosopher, logician, mathematician and scientist Charles Sanders Peirce (1839-1914). - _Amiram Eldar_, Jun 11 2021
%D A094577 Donald E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.5.
%H A094577 Alois P. Heinz, <a href="/A094577/b094577.txt">Table of n, a(n) for n = 0..288</a>
%H A094577 Charles Sanders Peirce, <a href="http://www.jstor.org/stable/2369442">On the Algebra of Logic</a>, American Journal of Mathematics, Vol. 3 (1880), pp. 15-57.
%F A094577 a(n) = Sum_{k=0..n} binomial(n,k)*Bell(2*n-k).
%F A094577 a(n) = Sum_{k=0..n} (-1)^k*binomial(n, k)*Bell(2*n-k+1).
%F A094577 a(n) = exp(-1)*Sum_{k>=0} (k(k+1))^n/k!. - _Benoit Cloitre_, Dec 30 2005
%F A094577 a(n) = Sum_{k=0..n} binomial(n,k)*Bell(n+k). - _Vaclav Kotesovec_, Jul 29 2022
%e A094577 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
%p A094577 seq(add(binomial(n, k)*(bell(n+k)), k=0..n), n=0..14); # _Zerinvary Lajos_, Dec 01 2006
%p A094577 # The objective of this implementation is efficiency.
%p A094577 # m -> [a(0), a(1), ..., a(m-1)] for m > 0.
%p A094577 A094577_list := proc(m)
%p A094577    local A, R, M, n, k, j;
%p A094577    M := m+m-1; A := array(1..M);
%p A094577    j := 1; R := 1; A[1] := 1;
%p A094577    for n from 2 to M do
%p A094577       A[n] := A[1];
%p A094577       for k from n by -1 to 2 do
%p A094577          A[k-1] := A[k-1] + A[k]
%p A094577       od;
%p A094577       if is(n,odd) then
%p A094577          j := j+1; R := R,A[j] fi
%p A094577    od;
%p A094577 [R] end:
%p A094577 A094577_list(100); # example call - _Peter Luschny_, Jan 17 2011
%t A094577 f[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}]; Array[f, 15, 0]
%o A094577 (Python)
%o A094577 # requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
%o A094577 from itertools import accumulate
%o A094577 A094577_list, blist, b = [1], [1], 1
%o A094577 for n in range(2,502):
%o A094577     blist = list(accumulate([b]+blist))
%o A094577     b = blist[-1]
%o A094577     blist = list(accumulate([b]+blist))
%o A094577     b = blist[-1]
%o A094577     A094577_list.append(blist[-n])
%o A094577 # _Chai Wah Wu_, Sep 02 2014, updated _Chai Wah Wu_, Sep 20 2014
%Y A094577 Cf. A094574, A020556, A216078.
%Y A094577 Main diagonal of array in A011971.
%K A094577 nonn
%O A094577 0,2
%A A094577 _Vladeta Jovovic_, May 12 2004