A005493 -id:A005493 - cp's OEIS frontend

A343672 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

1, 3, 19, 181, 2299, 36501, 695427, 15457709, 392672651, 11221959685, 356339728243, 12446649786429, 474273933636411, 19577992095770837, 870345573347448803, 41455153171478627533, 2106173029315813515883, 113694251997087087941925, 6498401704686168598548435, 392062852538564346207533789

Offset: 0

Ilya Gutkovskiy, Apr 25 2021

nonn

Cf. A000670, A005493, A006155, A032032, A343673, A343674.

a[0] = 1; a[n_] := a[n] = 2 n a[n - 1] + Sum[Binomial[n, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[1/(2 (1 - x) - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: 1 / (2 * (1 - x) - exp(x)).

a(n) ~ n! / (2*(1 + LambertW(exp(1)/2)) * (1 - LambertW(exp(1)/2))^(n+1)). - Vaclav Kotesovec, Jun 20 2022

A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 10, 15, 5, 9, 13, 20, 17, 27, 37, 52, 6, 11, 16, 25, 21, 34, 47, 67, 26, 43, 60, 87, 77, 114, 151, 203, 7, 13, 19, 30, 25, 41, 57, 82, 31, 52, 73, 107, 94, 141, 188, 255, 37, 63, 89, 132, 115, 175, 235, 322, 141, 218, 295, 409, 372, 523, 674

Offset: 0

Mikhail Kurkov, Aug 23 2021 [verification needed]

Modulo 2 binomial transform of A243499(n).

Antti Karttunen, Table of n, a(n) for n = 0..8191
Kevin Ryde, PARI/GP Code

Cf. A000110, A000120, A002720, A005493, A005494, A007814, A008277, A035009, A045379, A053645, A129760, A143494, A143495, A196834, A265649, A295989.

Similar recurrences: A124758, A243499, A284005, A329369, A341392.

function a = A347204(max_n)
    a(1) = 1;
    a(2) = 2;
    for nloop = 3:max_n
        n = nloop-1;
        s = 0;
        for k = 0:floor(log2(n))-1
            s = s + a(1+A053645(n)-2^k*(mod(floor(n/(2^k)),2)));
        end
        a(nloop) = 2*a(A053645(n)+1) + s;
    end
end
function a_n = A053645(n)
    a_n = n - 2^floor(log2(n));
end % Thomas Scheuerle, Oct 25 2021

f[n_] := BitAnd[n, n - 1]; a[0] = 1; a[n_] := a[n] = a[f[n]/2] + a[Floor[(n + f[n])/2]]; Array[a, 100, 0] (* Amiram Eldar, Nov 19 2021 *)

f(n) = bitand(n, n-1); \\ A129760
a(n) = if (n<=1, n+1, if (n%2, a(n\2)+a(n-1), a(f(n/2)) + a(n/2+f(n/2)))); \\ Michel Marcus, Oct 25 2021

PARI
```
\\ Also see links.
```

A129760(n) = bitand(n, n-1);
memoA347204 = Map();
A347204(n) = if (n<=1, n+1, my(v); if(mapisdefined(memoA347204,n,&v), v, v = if(n%2, A347204(n\2)+A347204(n-1), A347204(A129760(n/2)) + A347204(n/2+A129760(n/2))); mapput(memoA347204,n,v); (v))); \\ (Memoized version of Michel Marcus's program given above) - Antti Karttunen, Nov 20 2021

a(n) = a(n - 2^f(n)) + (1 + f(n))*a((n - 2^f(n))/2) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(2n+1) = a(n) + a(2n) for n >= 0.
a(2n) = a(n - 2^f(n)) + a(2n - 2^f(n)) for n > 0 with a(0) = 1 where f(n) = A007814(n).
a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) - 2^k*T(n,k)) for n > 1 with a(0) = 1, a(1) = 2, and where f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2.
Sum_{k=0..2^n - 1} a(k) = A035009(n+1) for n >= 0.
a((4^n - 1)/3) = A002720(n) for n >= 0.
a(2^n - 1) = A000110(n+1),
a(2*(2^n - 1)) = A005493(n),
a(2^2*(2^n - 1)) = A005494(n),
a(2^3*(2^n - 1)) = A045379(n),
a(2^4*(2^n - 1)) = A196834(n),
a(2^m*(2^n-1)) = T(n,m+1) is the n-th (m+1)-Bell number for n >= 0, m >= 0 where T(n,m) = m*T(n-1,m) + Sum_{k=0..n-1} binomial(n-1,k)*T(k,m) with T(0,m) = 1.
a(n) = Sum_{j=0..2^A000120(n)-1} A243499(A295989(n,j)) for n >= 0. Also A243499(n) = Sum_{j=0..2^f(n)-1} (-1)^(f(n)-f(j)) a(A295989(n,j)) for n >= 0 where f(n) = A000120(n). In other words, a(n) = Sum_{j=0..n} (binomial(n,j) mod 2)*A243499(j) and A243499(n) = Sum_{j=0..n} (-1)^(f(n)-f(j))*(binomial(n,j) mod 2)*a(j) for n >= 0 where f(n) = A000120(n).
Generalization:
b(n, x) = (1/x)*b((n - 2^f(n))/2, x) + (-1)^n*b(floor((2n - 2^f(n))/2), x) for n > 0 with b(0, x) = 1 where f(n) = A007814(n).
Sum_{k=0..2^n - 1} b(k, x) = (1/x)^n for n >= 0.
b((4^n - 1)/3, x) = (1/x)^n*n!*L_{n}(x) for n >= 0 where L_{n}(x) is the n-th Laguerre polynomial.
b((8^n - 1)/7, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A265649(n, k) for n >= 0.
b(2^n - 1, x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A008277(n+1, k+1),
b(2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143494(n+2, k+2),
b(2^2*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*A143495(n+3, k+3),
b(2^m*(2^n - 1), x) = (1/x)^n*Sum_{k=0..n} (-x)^k*T(n+m+1, k+m+1, m+1) for n >= 0, m >= 0 where T(n,k,m) is m-Stirling numbers of the second kind.

A011967 4th differences of Bell numbers.

Original entry on oeis.org

4, 15, 67, 322, 1657, 9089, 52922, 325869, 2114719, 14418716, 103004851, 769052061, 5987339748, 48506099635, 408157244967, 3561086589202, 32164670915029, 300324194090773, 2894932531218482, 28773297907499129

Offset: 0

N. J. A. Sloane

nonn

Chai Wah Wu, Table of n, a(n) for n = 0..250
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841.
Cohn, Martin; Even, Shimon; Menger, Karl, Jr.; Hooper, Philip K.; On the Number of Partitionings of a Set of n Distinct Objects, Amer. Math. Monthly 69 (1962), no. 8, 782--785. MR1531841. [Annotated scanned copy]
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Cf. A000110, A005493, A011965, A011966, A106436.

Differences[BellB[Range[0, 50]], 4] (* Vladimir Joseph Stephan Orlovsky, May 25 2011 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011967_list, blist, b = [4], [5, 7, 10, 15], 15
for _ in range(250):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A011967_list.append(blist[-5]) # Chai Wah Wu, Sep 20 2014

A102287 Total number of even blocks in all partitions of n-set.

Original entry on oeis.org

0, 1, 3, 13, 55, 256, 1274, 6791, 38553, 232171, 1477355, 9898780, 69621864, 512585529, 3940556611, 31560327945, 262805569159, 2271094695388, 20333574916690, 188322882941471, 1801737999086129, 17783472151154007, 180866601699482803, 1893373126840572056

Offset: 1

Vladeta Jovovic, Feb 19 2005

			a(3)=3 because in the 5 (=A000110(3)) partitions 123, (12)/3, (13)/2, 1/(23) and 1/2/3 of {1,2,3} we have 3 blocks of even size (shown between parentheses).

Alois P. Heinz, Table of n, a(n) for n = 1..575

Cf. A005493, A000296.

G:=(cosh(x)-1)*exp(exp(x)-1): Gser:=series(G,x=0,28): seq(n!*coeff(Gser,x^n),n=1..25); # Emeric Deutsch, Jun 22 2005
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0],
       add((p->(p+[0, `if`(i::odd, 0, j)*p[1]]))(
       b(n-i*j, i-1))*multinomial(n, n-i*j, i$j)/j!, j=0..n/i))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 16 2015

Range[0, nn]! CoefficientList[
  D[Series[Exp[y (Cosh[x] - 1) + Sinh[x]], {x, 0, nn}], y] /. y -> 1,  x]  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f: (cosh(x)-1)*exp(exp(x)-1).

More terms from Emeric Deutsch, Jun 22 2005

A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 3, 5, 0, 1, 5, 10, 15, 0, 1, 9, 22, 37, 52, 0, 1, 17, 52, 99, 151, 203, 0, 1, 33, 130, 283, 471, 674, 877, 0, 1, 65, 340, 855, 1561, 2386, 3263, 4140, 0, 1, 129, 922, 2707, 5451, 8930, 12867, 17007, 21147, 0, 1, 257, 2572, 8919, 19921, 35098, 53411, 73681, 94828, 115975

Offset: 1

Christian G. Bower, Jun 03 2005; Emeric Deutsch, Nov 14 2006

Another way to obtain this sequence (with offset 0): Form the infinite array U(n,k) = number of labeled partitions of (n,k) into pairs (i,j), for n >= 0, k >= 0 and read it by antidiagonals. In other words, U(n,k) = number of partitions of n black objects labeled 1..n and k white objects labeled 1..k. Each block must have at least one white object.

Then T(n,k)=U(n+k,k+1). Thus the two versions are related like "multichoose" to "choose". - Augustine O. Munagi, Jul 16 2007

			Triangle T(n,k) starts:
  1;
  0,1;
  0,1,2;
  0,1,3,5;
  0,1,5,10,15;
T(5,3)=5 because we have 1245|3, 145|2|3, 14|25|3, 15|24|3 and 1|245|3.
The arrays U(n,k) starts:
   1  0  0   0   0 ...
   1  1  1   1   1 ...
   2  3  5   9  17 ...
   5 10 22  52 130 ...
  15 37 99 283 855 ...

Alois P. Heinz, Rows n = 1..141, flattened

Row sums of T(n, k) yield A124496(n, 1).
Cf. A108461.
Columns of U(n, k): A000110, A005493, A033452.
Rows of U(n, k): A000007, A000012, A000051.
Main diagonal: A108459.

T[n_, k_] := Sum[If[n == k, 1, i^(n-k)]*StirlingS2[k, i], {i, 0, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 10 2024, after Vladeta Jovovic *)

T(n,1)=0 for n>=2; T(n,2)=1 for n>=2; T(n,3)=1+2^(n-3) for n>=3; T(n,n)=B(n-1), T(n,n-1)=B(n-1)-B(n-2), where B(q) are the Bell numbers (A000110).
Double e.g.f.: exp(exp(x)*(exp(y)-1)).
U(n,k) = Sum_{i=0..k} i^(n-k)*Stirling2(k,i). - Vladeta Jovovic, Jul 12 2007

Edited by N. J. A. Sloane, May 22 2008, at the suggestion of Vladeta Jovovic. This entry is a composite of two entries submitted independently by Christian G. Bower and Emeric Deutsch, with additional comments from Augustine O. Munagi.

A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].

Original entry on oeis.org

0, 0, 1, 3, 13, 54, 262, 1294, 7109, 40367, 248651, 1587414, 10827740, 76494630, 571499993, 4414720825, 35798107309, 299547765240, 2616358573834, 23536296521084, 220030456297349, 2114721297588097, 21046291460160803, 214984439282684504, 2267305399918683232

Offset: 0

Alois P. Heinz, Jun 05 2023

nonn

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

Alois P. Heinz, Table of n, a(n) for n = 0..575
Wikipedia, Partition of a set

Cf. A005493, A124418, A124425, A138378, A363434.

b:= proc(n, x, y, m) option remember; `if`(n=0, m,
      `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);

a(n) = Sum_{k=0..floor(n/2)} k * A124418(n,k).

a(n) = A138378(n) - A363434(n) = A005493(n-1) - A363434(n) for n>=1.

A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).

Original entry on oeis.org

1, 5, 28, 173, 1165, 8468, 65923, 546197, 4791214, 44301143, 430158397, 4372004546, 46381674085, 512328076385, 5879362011436, 69958289731457, 861605015493073, 10965899141265500, 144018319806024991, 1949190279770578145, 27153595018237222774

Offset: 0

Ilya Gutkovskiy, Dec 04 2023

nonn

Cf. A000110, A005493, A035009, A078940, A355247, A367888.

nmax = 20; CoefficientList[Series[Exp[3 (Exp[x] - 1) + 2 x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
Table[Sum[Binomial[n, k] 2^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 20}]

my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) + 2*x))) \\ Michel Marcus, Dec 04 2023

G.f. A(x) satisfies: A(x) = 1 + x * ( 2 * A(x) + 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k+2)^n / k!.
a(0) = 1; a(n) = 2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n-k) * A027710(k).

A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

Original entry on oeis.org

0, 3, 28, 210, 1506, 10871, 80592, 618939, 4942070, 41076508, 355372524, 3198027157, 29905143464, 290243182755, 2920041395248, 30414515081650, 327567816748638, 3643600859114439, 41809197852127240, 494367554679088923, 6017481714095327410

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 2..510
T. Hearne and C. G. Wagner, Minimal covers of finite sets, Discr. Math. 5 (1973), 247-251.

Cf. A046165.

Column k=1 of A282575.

seq(n*add((2^k-k-1)*Stirling2(n-1,k),k=1..n-1), n = 2 .. 30); # Robert Israel, May 21 2015

nn = 20; Range[0, nn]! CoefficientList[Series[Sum[ (Exp[x] - 1)^n/n! (2^n - n - 1) x, {n, 0, nn}], {x, 0, nn}], x] (* Geoffrey Critzer, Feb 18 2017 *)
a[2]=0;a[3]=3;a[4]=28;a[n_]:=n*Sum[(2^k-k-1)* StirlingS2[n-1,k], {k,1,n-1}];Table[a[n],{n,2,22}] (* Indranil Ghosh, Feb 20 2017 *)

a(n) = n * Sum_{k=1..n-1} (2^k-k-1) * S2(n-1,k) where S2(n,k) are the Stirling numbers of the second kind. - Sean A. Irvine, May 20 2015

a(n) = n * (A001861(n-1) - A005493(n-2) - A000110(n-1)). - Robert Israel, May 21 2015

More terms from Sean A. Irvine, May 20 2015

Title clarified by Geoffrey Critzer, Feb 18 2017

A056861 Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have an increase at index k (1<=k

Original entry on oeis.org

1, 3, 2, 10, 7, 6, 37, 27, 23, 21, 151, 114, 97, 88, 83, 674, 523, 446, 403, 378, 363, 3263, 2589, 2217, 1999, 1867, 1785, 1733, 17007, 13744, 11829, 10658, 9923, 9452, 9145, 8942, 94828, 77821, 67340, 60689, 56380, 53541, 51644, 50361, 49484, 562595

Offset: 2

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

Number of rises s_{k+1} > s_k in an RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, and s_i <= 1 + max(j

Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. - Franklin T. Adams-Watters, Jun 08 2006

			For example, [1, 2, 1, 2, 2, 3] is the RGS of a set partition of {1, 2, 3, 4, 5, 6} and has 3 rises, at i = 1, i = 3 and i = 5.
1;
3,2;
10,7,6;
37,27,23,21;
151,114,97,88,83;
674,523,446,403,378,363;
3263,2589,2217,1999,1867,1785,1733;
17007,13744,11829,10658,9923,9452,9145,8942;
94828,77821,67340,60689,56380,53541,51644,50361,49484;
562595,467767,406953,367101,340551,322619,310365,301905,296011,291871;
3535027,2972432,2599493,2348182,2176575,2058068,1975425,1917290,1876075, 1846648,1825501;

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]

Alois P. Heinz, Rows n = 2..100, flattened

Cf. Bell numbers A005493, A011965.

Cf. A056857-A056863.

b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[jJean-François Alcover, May 23 2016, after Alois P. Heinz *)

Edited and extended by Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 08 2016
Several terms corrected, R. J. Mathar, Mar 08 2016

A056862 Triangle T(n,k) is the number of restricted growth strings (RGS) of set partitions of {1..n} that have a decrease at index k (1<=k

Original entry on oeis.org

0, 0, 1, 0, 3, 4, 0, 10, 14, 16, 0, 37, 54, 63, 68, 0, 151, 228, 271, 296, 311, 0, 674, 1046, 1264, 1396, 1478, 1530, 0, 3263, 5178, 6349, 7084, 7555, 7862, 8065, 0, 17007, 27488, 34139, 38448, 41287, 43184, 44467, 45344, 0, 94828, 155642, 195494, 222044, 239976, 252230, 260690, 266584, 270724

Offset: 2

Winston C. Yang (winston(AT)cs.wisc.edu), Aug 31 2000

Number of falls s_k > s_{k+1} in a RGS [s_1, ..., s_n] of a set partition of {1, ..., n}, where s_i is the subset containing i, s_1 = 1 and s_i <= 1 + max(j

Note that the number of equalities at any index is B(n-1), where B(n) are the Bell numbers. - Franklin T. Adams-Watters, Jun 08 2006

			For example, [1, 2, 1, 2, 2, 3] is the RGS of a set partition of {1, 2, 3, 4, 5, 6} and has 1 fall, at i = 2.
0;
0,1;
0,3,4;
0,10,14,16;
0,37,54,63,68;
0,151,228,271,296,311;
0,674,1046,1264,1396,1478,1530;
0,3263,5178,6349,7084,7555,7862,8065;
0,17007,27488,34139,38448,41287,43184,44467,45344;
0,94828,155642,195494,222044,239976,252230,260690,266584,270724;
0,562595,935534,1186845,1358452,1476959,1559602,1617737,1658952,1688379, 1709526;

W. C. Yang, Conjectures on some sequences involving set partitions and Bell numbers, preprint, 2000. [apparently unpublished, Joerg Arndt, Mar 05 2016]

Alois P. Heinz, Rows n = 2..100, flattened

Cf. Bell numbers A005493.

Cf. A056857-A056863.

b:= proc(n, i, m, t) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, `if`(j (p-> seq(coeff(p, x, i), i=1..n-1))(b(n, 1, 0$2)[2]):
seq(T(n), n=2..12);  # Alois P. Heinz, Mar 24 2016

b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, If[jJean-François Alcover, May 23 2016, after Alois P. Heinz *)

T(n,k) = B(n) - B(n-1) - A056861(n,k). - Franklin T. Adams-Watters, Jun 08 2006

Conjecture: T(n,3) = 2*A011965(n). - R. J. Mathar, Mar 08 2016

Edited and extended by Franklin T. Adams-Watters, Jun 08 2006
Clarified definition and edited comment and example, Joerg Arndt, Mar 08 2016
Data corrected, R. J. Mathar, Mar 08 2016

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cp's OEIS Frontend

A343672 a(0) = 1; a(n) = 2 * n * a(n-1) + Sum_{k=0..n-1} binomial(n,k) * a(k).

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A347204 a(n) = a(f(n)/2) + a(floor((n+f(n))/2)) for n > 0 with a(0) = 1 where f(n) = A129760(n).

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A011967 4th differences of Bell numbers.

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A102287 Total number of even blocks in all partitions of n-set.

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A108458 Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the last block is the singleton {k}, 1<=k<=n; the blocks are ordered with increasing least elements.

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A362495 Total number of blocks containing at least one odd element and at least one even element in all partitions of [n].

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A367889 Expansion of e.g.f. exp(3*(exp(x) - 1) + 2*x).

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A003466 Number of minimal covers of an n-set that have exactly one point which appears in more than one set in the cover.

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A367889 Expansion of e.g.f. exp(3(exp(x) - 1) + 2x).