A094577 -id:A094577 - cp's OEIS frontend

A011971 Aitken's array: triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} read by rows, defined by a(0,0)=1, a(n,0) = a(n-1,n-1), a(n,k) = a(n,k-1) + a(n-1,k-1).

1, 1, 2, 2, 3, 5, 5, 7, 10, 15, 15, 20, 27, 37, 52, 52, 67, 87, 114, 151, 203, 203, 255, 322, 409, 523, 674, 877, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147, 21147, 25287, 30304, 36401, 43833, 52922, 64077, 77821, 94828, 115975

Offset: 0

N. J. A. Sloane, J. H. Conway and R. K. Guy

Also called the Bell triangle or the Peirce triangle.
a(n,k) is the number of equivalence relations on {0, ..., n} such that k is not equivalent to n, k+1 is not equivalent to n, ..., n-1 is not equivalent to n. - Don Knuth, Sep 21 2002 [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008]
Named after the New Zealand mathematician Alexander Craig Aitken (1895-1967). - Amiram Eldar, Jun 11 2021

			Triangle begins:
00:       1
01:       1      2
02:       2      3      5
03:       5      7     10     15
04:      15     20     27     37     52
05:      52     67     87    114    151    203
06:     203    255    322    409    523    674    877
07:     877   1080   1335   1657   2066   2589   3263   4140
08:    4140   5017   6097   7432   9089  11155  13744  17007  21147
09:   21147  25287  30304  36401  43833  52922  64077  77821  94828 115975
10:  115975 137122 162409 192713 229114 272947 325869 389946 467767 562595 678570
...

Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 205.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 212.
Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).
Charles Sanders Peirce, On the Algebra of Logic, American Journal of Mathematics, Vol. 3, pages 15-57, 1880. Reprinted in Collected Papers (1935-1958) and in Writings of Charles S. Peirce: A Chronological Edition (Indiana University Press, Bloomington, IN, 1986).
Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.

T. D. Noe and Chai Wah Wu, Rows n = 0..200 of triangle, flattened (rows n = 0..50 from T. D. Noe)
Alexander Craig Aitken, A problem in combinations, Edinburgh Mathematical Notes, Vol. 28 (1933), pp. xviii-xxiii.
H. W. Becker, Rooks and rhymes, Math. Mag., Vol. 22, No. 1 (1948/49), pp. 23-26. See Table IV.
H. W. Becker, Rooks and rhymes, Math. Mag., Vol. 22, No. 1 (1948/49), pp. 23-26. [Annotated scanned copy]
Antonio Bernini, Mathilde Bouvel and Luca Ferrari, Some statistics on permutations avoiding generalized patterns, PU.M.A. Vol. 18, No. 3-4 (2007), pp. 223-237 (see array p. 228).
Clarence H. Best, Jerry Griggs, and Ira Gessel, Partitions of finite sets, Advanced Problem 6151, The American Mathematical Monthly, Vol. 86, No. 1 (Jan., 1979), pp. 64-65.
Robert W. Donley, Jr., Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019.
Martin Cohn, Shimon Even, Karl Menger, Jr. and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly, Vol. 69, No. 8 (1962), pp. 782-785. MR1531841.
Martin Cohn, Shimon Even, Karl Menger, Jr. and Philip K. Hooper, On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly Vol. 69, No. 8 (1962), pp. 782-785. MR1531841. [Annotated scanned copy]
Dominique Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.
Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.
Nick Hobson, Python program for this sequence.
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018.
Charles Sanders Peirce, Assorted Papers.
Charles Sanders Peirce, Collected Papers.
Charles Sanders Peirce, Published Works.
Jeffrey Shallit, A triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
Todd Tichenor, Bounds on graph compositions and the connection to the Bell triangle, Discr. Math., Vol. 339, No. 4 (2016), pp. 1419-1423.
Eric Weisstein's World of Mathematics, Bell Triangle.
Denys Wuilquin, Letters to N. J. A. Sloane, August 1984.

Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, etc., A011968, A011969. Cf. A046934, A011972 (duplicates removed).
Main diagonal is in A094577. Mirror image is in A123346.
See also A095149, A106436, A108041, A108042, A108043.

T:=Flat(List([0..9],n->List([0..n],k->Sum([0..k],i->Binomial(k,i)*Bell(n-k+i))))); # Muniru A Asiru, Oct 26 2018

a011971 n k = a011971_tabl !! n !! k
a011971_row n = a011971_tabl !! n
a011971_tabl = iterate (\row -> scanl (+) (last row) row) [1]
-- Reinhard Zumkeller, Dec 09 2012

A011971 := proc(n,k) option remember; if n=0 and k=0 then 1 elif k=0 then A011971(n-1,n-1) else A011971(n,k-1)+A011971(n-1,k-1); fi: end;
for n from 0 to 12 do lprint([ seq(A011971(n,k),k=0..n) ]); od:
# Compare the analogue algorithm for the Catalan numbers in A030237.
BellTriangle := proc(len) local P, T, n; P := [1]; T := [[1]];
for n from 1 to len - 1 do P := ListTools:-PartialSums([P[-1], op(P)]);
T := [op(T), P] od; T end:
BellTriangle(6); ListTools:-Flatten(%); # Peter Luschny, Mar 26 2022

a[0, 0] = 1; a[n_, 0] := a[n, 0] = a[n-1, n-1]; a[n_, k_] := a[n, k] = a[n, k-1] + a[n-1, k-1]; Flatten[ Table[ a[n, k], {n, 0, 9}, {k, 0, n}]] (* Robert G. Wilson v, Mar 27 2004 *)
Flatten[Table[Sum[Binomial[k, i]*BellB[n-k+i], {i, 0, k}], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, May 24 2016, after Vladeta Jovovic *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011971 = blist = [1]
for _ in range(10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A011971 += blist # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 19 2014

Double-exponential generating function: Sum_{n, k} a(n-k, k) x^n y^k / n! k! = exp(e^{x+y}-1+x). - Don Knuth, Sep 21 2002 [U coordinates, reversed]

a(n,k) = Sum_{i=0..k} binomial(k,i)*Bell(n-k+i). - Vladeta Jovovic, Oct 15 2006

Peirce reference from Jon Awbrey, Mar 11 2002
Reference to my paper from Jeffrey Shallit, Jan 23 2015
Moved a comment to A056857 where it belonged. - N. J. A. Sloane, May 02 2015.

A094574 Number of (<=2)-covers of an n-set.

Original entry on oeis.org

1, 1, 5, 40, 457, 6995, 136771, 3299218, 95668354, 3268445951, 129468914524, 5868774803537, 301122189141524, 17327463910351045, 1109375488487304027, 78484513540137938209, 6098627708074641312182, 517736625823888411991202, 47791900951140948275632148

Offset: 0

Goran Kilibarda, Vladeta Jovovic, May 12 2004

nonn

Also the number of strict multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. For example, the a(2) = 5 strict multiset partitions of {1, 1, 2, 2} are (1122), (1)(122), (2)(112), (11)(22), (1)(2)(12). - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Sep 02 2019: (Start)
These are set-systems covering {1..n} with vertex-degrees <= 2. For example, the a(3) = 40 covers are:
  {123}  {1}{23}    {1}{2}{3}     {1}{2}{3}{12}
         {2}{13}    {1}{2}{13}    {1}{2}{3}{13}
         {3}{12}    {1}{2}{23}    {1}{2}{3}{23}
         {1}{123}   {1}{3}{12}    {1}{2}{13}{23}
         {12}{13}   {1}{3}{23}    {1}{2}{3}{123}
         {12}{23}   {2}{3}{12}    {1}{3}{12}{23}
         {13}{23}   {2}{3}{13}    {2}{3}{12}{13}
         {2}{123}   {1}{12}{23}
         {3}{123}   {1}{13}{23}
         {12}{123}  {1}{2}{123}
         {13}{123}  {1}{3}{123}
         {23}{123}  {2}{12}{13}
                    {2}{13}{23}
                    {2}{3}{123}
                    {3}{12}{13}
                    {3}{12}{23}
                    {12}{13}{23}
                    {1}{23}{123}
                    {2}{13}{123}
                    {3}{12}{123}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A219585. - Alois P. Heinz, Nov 23 2012
Dominated by A003465.
Graphs with vertex-degrees <= 2 are A136281.
Cf. A002718, A007716, A020554, A020555, A050535, A094574, A136284, A316974, A327104, A327106, A327229.
Main diagonal of A346517.

facs[n_]:=facs[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[Array[Prime,n,1,Times]^2],UnsameQ@@#&]],{n,0,6}] (* Gus Wiseman, Jul 18 2018 *)
m = 20;
a094577[n_] := Sum[Binomial[n, k]*BellB[2 n - k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
CoefficientList[egf + O[x]^m, x]*Range[0, m-1]! (* Jean-François Alcover, May 13 2019 *)

Row sums of A094573.

E.g.f: exp(-1-1/2*(exp(x)-1))*Sum(exp(x*binomial(n+1, 2))/n!, n=0..infinity) or exp((1-exp(x))/2)*Sum(A094577 (n)*(x/2)^n/n!, n=0..infinity).

A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

Original entry on oeis.org

1, 1, 3, 7, 27, 87, 409, 1657, 9089, 43833, 272947, 1515903, 10515147, 65766991, 501178937, 3473600465, 28773452321, 218310229201, 1949230218691, 16035686850327, 153281759047387, 1356791248984295, 13806215066685433, 130660110400259849, 1408621900803060705

Offset: 1

R. H. Hardin, Sep 01 2012

nonn

Number of vertex covers and independent vertex sets of the n-1 X n-1 white bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the white squares of an n-1 X n-1 board. - Andrew Howroyd, May 08 2017

Number of pairs of partitions (A<=B) of [n-1] such that the nontrivial blocks of A are of type {k,n-1-k} if n is even, and of type {k,n-k} if n is odd. - Francesca Aicardi, May 28 2022

			Some solutions for n = 5:
  x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0
  1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x
  x 2   x 0   x 0   x 2   x 0   x 1   x 1   x 2   x 2   x 1
  0 x   2 x   1 x   3 x   1 x   0 x   2 x   3 x   0 x   0 x
  x 3   x 1   x 2   x 2   x 0   x 1   x 1   x 1   x 2   x 0
There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - _Andrew Howroyd_, Jun 06 2017

R. H. Hardin, Table of n, a(n) for n = 1..210
Eric Weisstein's World of Mathematics, Independent Vertex Set
Eric Weisstein's World of Mathematics, Vertex Cover
Eric Weisstein's World of Mathematics, White Bishop Graph

Column 2 of A216084.
Row sums of A274106(n-1).
Cf. A020556, A094577, A216332, A201862, A286423.

a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e)
           , k=0..m))(iquo(n-1, 2, 'e')):
seq(a(n), n=1..26);  # Alois P. Heinz, Oct 03 2022

a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2];
   Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 25 2022, after Francesca Aicardi *)

a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - Francesca Aicardi, May 28 2022
From Vaclav Kotesovec, Jul 29 2022: (Start)
a(2*k) = A020556(k).
a(2*k+1) = A094577(k). (End)

A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 15, 10, 7, 5, 52, 37, 27, 20, 15, 203, 151, 114, 87, 67, 52, 877, 674, 523, 409, 322, 255, 203, 4140, 3263, 2589, 2066, 1657, 1335, 1080, 877, 21147, 17007, 13744, 11155, 9089, 7432, 6097, 5017, 4140, 115975, 94828, 77821, 64077, 52922, 43833, 36401, 30304, 25287, 21147

Offset: 0

N. J. A. Sloane, Oct 14 2006

a(n,k) is k-th difference of Bell numbers, with a(n,1) = A000110(n) for  n>0, a(n,k) = a(n,k-1) - a(n-1, k-1), k<=n, with diagonal (k=n) also equal to Bell numbers (n>=0). - Richard R. Forberg, Jul 13 2013
From Don Knuth, Jan 29 2018: (Start)
If the offset here is changed from 0 to 1, then we can say:
a(n,k) is the number of equivalence classes of [n] in which 1 not equiv to 2, ..., 1 not equiv to k.
In Volume 4A, page 418, I pointed out that a(n,k) is the number of set partitions in which k is the smallest of its block.
And in exercise 7.2.1.5--33, I pointed out that a(n,k) is the number of equivalence relations in which 1 not equiv to 2, 2 not equiv to 3, ..., k-1 not equiv to k. (End)

			Triangle begins:
    1
    2   1
    5   3   2
   15  10   7  5
   52  37  27 20 15
  203 151 114 87 67 52
  ...

D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 418).

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened
A. Dil, Veli Kurt, Investigating Geometric and Exponential Polynomials with Euler-Seidel Matrices, J. Int. Seq. 14 (2011) # 11.4.6
Don Knuth, Email to N. J. A. Sloane, Jan 29 2018
Eric Weisstein's World of Mathematics, Bell Triangle.

Cf. A011971. Borders give Bell numbers A000110. Diagonals give A005493, A011965, A011966, A011968, A011969, A046934, A011972, A094577, A095149, A106436, A108041, A108042, A108043.

a123346 n k = a123346_tabl !! n !! k
a123346_row n = a123346_tabl !! n
a123346_tabl = map reverse a011971_tabl
-- Reinhard Zumkeller, Dec 09 2012

a[n_, k_] := Sum[Binomial[n - k, i - k] BellB[i], {i, k, n}];
Table[a[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2018 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A123346_list = blist = [1]
for _ in range(2*10**2):
    b = blist[-1]
    blist = list(accumulate([b]+blist))
    A123346_list += reversed(blist)
# Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 20 2014

a(n,k) = Sum_{i=k..n} binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

More terms from Alexander Adamchuk and Vladeta Jovovic, Oct 14 2006

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 27, 13, 1, 1, 15, 409, 778, 73, 1, 1, 52, 9089, 104149, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset: 0

Peter Luschny, Mar 29 2011

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
The square array starts for n>=0, k>=0:
n\k=0,1,..  A000012,A000262,A182934,...
A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
       : 1, 73, 37553, 57184313, 192052025697, ...
       : 1, 501, 2688546, 56410245661, ...
 ....  : 1, 4051, 265141267, 89501806774945, ...

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182925, A182924, A182933.

A182933_AsSquareArray := proc(n,k) local r,s,i;
r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);

a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).

Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).

A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

Original entry on oeis.org

1, 2, 8, 59, 652, 9736, 186478, 4421018, 126317785, 4260664251, 166884941780, 7489637988545, 380861594219460, 21739310882945458, 1381634777325000263, 97089956842985393297, 7497783115765911443879, 632884743974716421132084

Offset: 0

Daniel E. Loeb, Dec 16 2010

nonn

If each element must appear in exactly 1 subset, then we get the Bell numbers A000110.

If each element must appear in exactly 2 subsets, then we get A002718.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Row n=2 of A330964.

Cf. A094574, A000110, A002718, A178171, A178173.

terms = m = 30;
a094577[n_] := Sum[Binomial[n, k]*BellB[2n-k], {k, 0, n}];
egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
A094574 = CoefficientList[egf + O[x]^m, x]*Range[0, m-1]!;
a[n_] := Sum[Binomial[n, k]*A094574[[k+1]], {k, 0, n}];
Table[a[n], {n, 0, m-1}] (* Jean-François Alcover, May 24 2019 *)

from numpy import array
def toBinary(n, k):
    ans=[]
    for i in range(k):
        ans.insert(0, n%2)
        n=n>>1
    return array(ans)
def powerSet(k): return [toBinary(n,k) for n in range(1,2**k)]
def courcelle(maxUses, remainingSets, exact=False):
    if exact and not all(maxUses<=sum(remainingSets)): ans=0
    elif len(remainingSets)==0: ans=1
    else:
        set0=remainingSets[0]
        if all(set0<=maxUses): ans=courcelle(maxUses-set0,remainingSets[1:],exact=exact)
        else: ans=0
        ans+=courcelle(maxUses,remainingSets[1:],exact=exact)
    return ans
for i in range(10):
    print(i, courcelle(array([2]*i),powerSet(i),exact=False))

Binomial transform of A094574: a(n) = Sum_{k=0..n} C(n,k)*A094574(k).

Edited and corrected by Max Alekseyev, Dec 19 2010

A363452 Total number of blocks containing only odd elements in all partitions of [n].

Original entry on oeis.org

0, 1, 1, 5, 12, 62, 206, 1189, 4949, 31775, 156972, 1110280, 6301550, 48637701, 310279615, 2591820857, 18293310174, 164218811718, 1267153412532, 12152174863961, 101557600812015, 1035203191874931, 9299499328238110, 100314319611860936, 962663031508255416

Offset: 0

Alois P. Heinz, Jun 02 2023

nonn

			a(3) = 5 = 0 + 1 + 1 + 1 + 2 : 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Partition of a set

Cf. A000110, A094577, A124420, A363434, A363453.

b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(u, i)*
      add(Stirling2(g, j)*j^(u-i), j=0..g), i=k..u)
    end:
a:= n-> add(b(n, k)*k, k=0..ceil(n/2)):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, x, y, m) option remember; `if`(n=0, y,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, y,
     If[x + m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
     If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..ceiling(n/2)} k * A124420(n,k).
a(n) = A363434(n) - A363453(n).
a(2n) = A363453(2n).
a(2n+1) = A363453(2n+1) + A094577(n).

A363453 Total number of blocks containing only even elements in all partitions of [n].

Original entry on oeis.org

0, 0, 1, 2, 12, 35, 206, 780, 4949, 22686, 156972, 837333, 6301550, 38122554, 310279615, 2090641920, 18293310174, 135445359397, 1267153412532, 10202944645270, 101557600812015, 881921432827544, 9299499328238110, 86508104545175503, 962663031508255416

Offset: 0

Alois P. Heinz, Jun 02 2023

nonn

			a(3) = 2 = 0 + 0 + 1 + 0 + 1 : 123, 12|3, 13|2, 1|23, 1|2|3.

Alois P. Heinz, Table of n, a(n) for n = 0..250
Wikipedia, Partition of a set

Cf. A000110, A094577, A124422, A363434, A363452.

b:= proc(n, k) local g, u; g:= floor(n/2); u:=ceil(n/2);
      add(Stirling2(i, k)*binomial(g, i)*
      add(Stirling2(u, j)*j^(g-i), j=0..u), i=k..g)
    end:
a:= n-> add(b(n, k)*k, k=0..floor(n/2)):
seq(a(n), n=0..25);
# second Maple program:
b:= proc(n, x, y, m) option remember; `if`(n=0, x,
      `if`(x+m>0, b(n-1, y, x, m)*(x+m), 0)+b(n-1, y, x+1, m)+
      `if`(y>0, b(n-1, y-1, x, m+1)*y, 0))
    end:
a:= n-> b(n, 0$3):
seq(a(n), n=0..25);

b[n_, x_, y_, m_] := b[n, x, y, m] = If[n == 0, x,
    If[x+m > 0, b[n-1, y, x, m]*(x+m), 0] + b[n-1, y, x+1, m] +
    If[y > 0, b[n-1, y-1, x, m+1]*y, 0]];
a[n_] := b[n, 0, 0, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

a(n) = Sum_{k=0..floor(n/2)} k * A124422(n,k).
a(n) = A363434(n) - A363452(n).
a(2n) = A363452(2n).
a(2n+1) = A363452(2n+1) - A094577(n).

cp's OEIS Frontend

A011971 Aitken's array: triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} read by rows, defined by a(0,0)=1, a(n,0) = a(n-1,n-1), a(n,k) = a(n,k-1) + a(n-1,k-1).

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A094574 Number of (<=2)-covers of an n-set.

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A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

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A123346 Mirror image of the Bell triangle A011971, which is also called the Pierce triangle or Aitken's array.

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A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

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A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

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A363452 Total number of blocks containing only odd elements in all partitions of [n].

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