A005493 -id:A005493 - cp's OEIS frontend

A138378 Number of embedded coalitions in an n-person game.

1, 3, 10, 37, 151, 674, 3263, 17007, 94828, 562595, 3535027, 23430840, 163254885, 1192059223, 9097183602, 72384727657, 599211936355, 5150665398898, 45891416030315, 423145657921379, 4031845922290572, 39645290116637023, 401806863439720943, 4192631462935194064

Offset: 1

David Yeung (wkyeung(AT)hkbu.edu.hk), May 08 2008

nonn

Same as A005493, apart from offset. - R. J. Mathar, Sep 23 2011

The strategic behavior of players depends crucially on the coalition structures of a game.

			a(1) = combination(1,0) = 1,
a(2) = combination(2,1) + combination(2,0)= 3,
a(3) = combination(3,2)* a(1) + combination(3,2) + combination(3,1) + combination(3,0)= 10,
a(4) = combination(4,3)* {a(1) + a(2)} + combination(4,2)* a(1) + combination7(4,3)combination(4,2) + combination(4,1) + combination(4,0)= 37,
a(5) = combination(5,4)* {a(1) + a(2) + a(3)} combination(5,3)* {a(1) + a(2)} + combination(5,2)* a(1) + combination(5,4) + combination(5,3) + combination(5,2) + combination(5,1) + combination(5,0)= 151.

J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag, New York, 1995.

Alois P. Heinz, Table of n, a(n) for n = 1..575
E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419.
Gábor Czédli, Lattices with lots of congruence energy, arXiv:2205.02294 [math.RA], 2022.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
David W. K. Yeung, Recursive sequence identifying the number of embedded coalitions, International Game Theory Review 10(2008), 129-136.

Cf. A000110, A005493, A048993, A138379.

Column k=1 of A283424.

[&+[k*StirlingSecond(n, k): k in [1..n]]: n in [1..25]]; // Vincenzo Librandi, May 18 2019

b:= proc(n, m) option remember;
     `if`(n=0, m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Dec 10 2024

a[n_] := Sum[Binomial[n, j] BellB[j], {j, 0, n-1}];
Array[a, 24] (* Jean-François Alcover, Aug 19 2018 *)

a(1) = combination(1,0) = 1, a(2) = combination(2,1) + combination(2,0)= 3, a(n) = {SUM(i=2 to n-1) combination(n,i)} * {SUM(j=1 to i-1) a(n)} + SUM(i=0 to n-1) combination(n,i), for n > 2.
G.f.: 1/U(0) where U(k)= 1 - x*(k+3) - x^2*(k+1)/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 11 2012
G.f.: 1/(U(0)-x) where U(k)= 1 - x - x*(k+1)/(1 - x/U(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 12 2012
G.f.: -G(0)/x^2 where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 08 2013
G.f.: Q(0)/x^2 -1/x^2, where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 10 2013
a(n) = Bell(n+1)-Bell(n) = Sum_{k=1..n} k*Stirling2(n,k). - Alois P. Heinz, May 11 2017
E.g.f.: (exp(x) - 1) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jan 26 2020

A132963 Total number of distinct block sizes in all partitions of [n].

Original entry on oeis.org

1, 2, 8, 25, 102, 439, 2067, 10406, 56754, 328257, 2015818, 13067366, 89192170, 638321285, 4779442602, 37332643831, 303635437532, 2565592977205, 22483754207839, 204013083946460, 1913880812797792, 18536832515581167, 185130415180288134, 1904280138346826637

Offset: 1

Vladeta Jovovic, Sep 06 2007

G. C. Greubel, Table of n, a(n) for n = 1..500

Cf. A005493, A132958, A132959, A132960, A132961, A132962, A350175.

b:= proc(n, i, c) option remember; `if`(n=0, c,
      `if`(i<1, 0, add(b(n-j*i, i-1, c+signum(j))*
      combinat[multinomial](n, n-i*j, i$j)/j!, j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Jan 06 2022

Rest[ Range[0, 23]! CoefficientList[ Series[ Exp[ Exp[x] - 1] Sum[1 - Exp[ -x^k/k! ], {k, 30}], {x, 0, 23}], x]] (* Robert G. Wilson v, Sep 13 2007 *)

E.g.f.: exp(exp(x)-1)*Sum_{k>0} (1-exp(-x^k/k!)).

More terms from Robert G. Wilson v, Sep 13 2007

A137650 Triangle read by rows, A008277 * A000012.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 14, 7, 1, 52, 51, 36, 11, 1, 203, 202, 171, 81, 16, 1, 877, 876, 813, 512, 162, 22, 1, 4140, 4139, 4012, 3046, 1345, 295, 29, 1, 21147, 21146, 20891, 17866, 10096, 3145, 499, 37, 1, 115975, 115974, 115463

Offset: 1

Gary W. Adamson, Feb 01 2008

Left column = Bell numbers (A000110) starting (1, 2, 5, 15, 52, 203, ...). Row sums = A005493(n+1): (1, 3, 10, 37, 151, 674, ...).

Corresponding to the generalized Stirling number triangle of first kind A049444. - Peter Luschny, Sep 18 2011

			First few rows of the triangle are
    1;
    2,   1;
    5,   4,   1;
   15,  14,   7,   1;
   52,  51,  36,  11,   1;
  203, 202, 171,  81,  16,   1;
  877, 876, 813, 512, 162,  22,   1;
  ...

Cf. A000110, A008277, A005493, A049444.

A similar triangle is A133611.

A137650_row := proc(n) local k,i;
add(add(combinat[stirling2](n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=0..n-1) end:
seq(print(A137650_row(n)),n=1..7); # Peter Luschny, Sep 18 2011

row[n_] := Table[StirlingS2[n, k], {k, 0, n}] // Reverse // Accumulate // Reverse // Rest;
Array[row, 10] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

A008277 * A000012 as infinite lower triangular matrices. Partial sums of A008277 rows starting from the right.

A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Original entry on oeis.org

1, 6, 37, 235, 1540, 10427, 73013, 529032, 3967195, 30785747, 247126450, 2050937445, 17585497797, 155666739742, 1421428484337, 13377704321695, 129659127547372, 1293095848212799, 13259069937250169, 139671750579429512, 1510382932875294447, 16754464511605466311

Offset: 0

Wolfdieter Lang, Oct 07 2011

			a(2) = 25 + 11 + 1 = 37.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A005494, A045379, A196835 (alternating row sums).

b:= proc(n, m) option remember;
     `if`(n=0, 1, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 5):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 22 2021

nmax = 20; CoefficientList[Series[E^(E^x + 5*x - 1), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 10 2020 *)

a(n) = Sum_{m=0..n} A193685(n,m).
E.g.f.: exp(exp(x)+5*x-1).
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n + 5) / LambertW(n)^(n + 11/2). - Vaclav Kotesovec, Jun 10 2020
a(0) = 1; a(n) = 5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Jul 03 2020

A190823 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 2.

Original entry on oeis.org

1, 0, 0, 1, 10, 99, 1146, 15422, 237135, 4106680, 79154927, 1681383864, 39034539488, 983466451011, 26728184505750, 779476074425297, 24281301468714902, 804688068731837874, 28269541494090294129, 1049450257149017422000, 41050171013933837206545

Offset: 0

R. H. Hardin, May 21 2011

From Gus Wiseman, Feb 27 2019: (Start)
Also the number of 2-uniform set partitions of {1..2n} such that no block has its two vertices differing by less than 3. For example, the a(4) = 10 set partitions are:
  {{1,4}, {2,6}, {3,7}, {5,8}}
  {{1,4}, {2,7}, {3,6}, {5,8}}
  {{1,5}, {2,6}, {3,7}, {4,8}}
  {{1,5}, {2,6}, {3,8}, {4,7}}
  {{1,5}, {2,7}, {3,6}, {4,8}}
  {{1,5}, {2,8}, {3,6}, {4,7}}
  {{1,6}, {2,5}, {3,7}, {4,8}}
  {{1,6}, {2,5}, {3,8}, {4,7}}
  {{1,7}, {2,5}, {3,6}, {4,8}}
  {{1,8}, {2,5}, {3,6}, {4,7}}
(End)

			All solutions for n=4 (read downwards):
  1    1    1    1    1    1    1    1    1    1
  2    2    2    2    2    2    2    2    2    2
  3    3    3    3    3    3    3    3    3    3
  4    4    4    4    1    4    4    1    4    4
  1    1    2    1    4    2    1    4    2    2
  3    3    1    2    2    3    2    3    1    3
  2    4    4    4    3    4    3    2    3    1
  4    2    3    3    4    1    4    4    4    4

Alois P. Heinz, Table of n, a(n) for n = 0..404
Dmitry Efimov, Hafnian of two-parameter matrices, arXiv:2101.09722 [math.CO], 2021.
Everett Sullivan, Linear chord diagrams with long chords, arXiv preprint arXiv:1611.02771 [math.CO], 2016.

Distance of 1 instead of 2 gives |A000806|.
Column k=3 of A293157.
Cf. A000699, A001147 (2-uniform set partitions), A003436, A005493, A011968, A170941, A278990 (distance 2+ version), A306386 (cyclical version).

I:=[1,0,0,1,10,99]; [n le 5 select I[n] else 2*n*Self(n-1) -2*(3*n-8)*Self(n-2) +2*(3*n-11)*Self(n-3) -2*(n-5)*Self(n-4) -Self(n-5): n in [1..40]]; // G. C. Greubel, Dec 03 2023

a[0]=1; a[1]=0; a[2]=0; a[3]=1; a[4]=10; a[5]=99; a[n_] := a[n] = (2*n+2) a[n-1] - (6*n-10) a[n-2] + (6*n-16) a[n-3] - (2*n-8) a[n-4] - a[n-5]; Array[a, 20, 0] (* based on Sullivan's formula, Giovanni Resta, Mar 20 2017 *)
dtui[{}]:={{}};dtui[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@dtui[Complement[set,s]]]/@Table[{i,j},{j,Select[set,#>i+2&]}];
Table[Length[dtui[Range[n]]],{n,0,12,2}] (* Gus Wiseman, Feb 27 2019 *)

@CachedFunction
def a(n): # a = A190823
    if (n<6): return (1,0,0,1,10,99)[n]
    else: return 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5)
[a(n) for n in range(41)] # G. C. Greubel, Dec 03 2023

a(n) = 2*(n+1)*a(n-1) - 2*(3*n-5)*a(n-2) + 2*(3*n-8)*a(n-3) - 2*(n-4)*a(n-4) - a(n-5) (proved). - Everett Sullivan, Mar 16 2017

a(n) ~ 2^(n+1/2) * n^n / exp(n+2), based on Sullivan's formula. - Vaclav Kotesovec, Mar 21 2017

a(16)-a(20) (using Everett Sullivan's formula) from Giovanni Resta, Mar 20 2017

a(0)=1 prepended by Alois P. Heinz, Oct 17 2017

A362924 Triangle read by rows: T(n,m), n >= 1, 1 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

Original entry on oeis.org

1, 2, 1, 5, 4, 1, 15, 13, 8, 1, 52, 47, 35, 16, 1, 203, 188, 153, 97, 32, 1, 877, 825, 706, 515, 275, 64, 1, 4140, 3937, 3479, 2744, 1785, 793, 128, 1, 21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1, 115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1, 678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1

Offset: 1

N. J. A. Sloane, Aug 10 2023, based on an email from Don Knuth

Also, the maximum number of solutions to an exact cover problem with n items, of which m are secondary.

			Triangle begins:
  [1],
  [2, 1],
  [5, 4, 1],
  [15, 13, 8, 1],
  [52, 47, 35, 16, 1],
  [203, 188, 153, 97, 32, 1],
  [877, 825, 706, 515, 275, 64, 1],
  [4140, 3937, 3479, 2744, 1785, 793, 128, 1],
  [21147, 20270, 18313, 15177, 11002, 6347, 2315, 256, 1],
  [115975, 111835, 102678, 88033, 68303, 45368, 23073, 6817, 512, 1],
  [678570, 657423, 610989, 536882, 436297, 316305, 191866, 85475, 20195, 1024, 1],
...
For example, if n=4, m=3, then T(4,3) = 8, because out of the A000110(4) = 15 set partitions of {1,2,3,4}, those that have 2 or more blocks contained in {1,2,3} are
  {12,3,4},
  {13,2,4},
  {14,2,3},
  {23,1,4},
  {24,1,3},
  {34,1,2},
  {1,2,3,4},
  while
  {1234},
  {123,4},
  {124,3}
  {134,2}
  {234,1},
  {12,34}
  {13. 24}.
  {14, 23}
  do not.

D. E. Knuth, The Art of Computer Programming, Volume 4B, exercise 7.2.2.1--185, answer on page 468.

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened).

See A113547 and A362925 for other versions of this triangle.
Row sums give A005493.
Cf. A000110, A008277, A078468, A143494.

with(combinat);
T:=proc(n,m) local k;
add(stirling2(n-m,k)*(k+1)^m, k=0..n-m);
end;

A362924[n_,m_]:=Sum[StirlingS2[n-m,k](k+1)^m,{k,0,n-m}];
Table[A362924[n,m],{n,15},{m,n}] (* Paolo Xausa, Dec 02 2023 *)

T(n, 1) = Bell number (all set partitions) A000110(n);
T(n, n) = 1 when m=n (the only possibility is a single block);
T(n, n-1) = 2^{n-1} when m=n-1 (a single block or two blocks);
T(n, 2) =  A078468(2).
In general, T(n, m) = Sum_{k=0..n-m} Stirling_2(n-m,k)*(k+1)^m.

A011966 Third differences of Bell numbers.

Original entry on oeis.org

1, 5, 20, 87, 409, 2066, 11155, 64077, 389946, 2504665, 16923381, 119928232, 888980293, 6876320041, 55382419676, 463539664643, 4024626253845, 36189297168874, 336513491259647, 3231446022478129, 32004743929977258, 326548129128737469, 3428663026172389201

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n+4 with at least one singleton and with the smallest element in a singleton equal to 4. Alternatively, number of partitions of n+4 with at least one singleton and with the largest element in a singleton equal to n+1. - Olivier GERARD, Oct 29 2007

Olivier Gérard and Karol A. Penson, A budget of set partition statistics, in preparation, unpublished as of Sep 22 2011.

Alois P. Heinz, Table of n, a(n) for n = 0..215
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. A106436, A011965.

a:= n-> add((-1)^(k+1)*binomial(3,k)*combinat['bell'](n+k), k=0..3):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 05 2008

Differences[BellB[Range[0,30]],3]  (* Harvey P. Dale, Apr 21 2011 *)

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A011966_list, blist, b = [1], [2, 3, 5], 5
for _ in range(1000):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A011966_list.append(blist[-4]) # Chai Wah Wu, Sep 20 2014

G.f.: -(1-x+x^2)/x^2 + (1-x)^3/x^2/(G(0)-x) where G(k) = 1 - x*(k+1)/(1 - x/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 26 2013
From Vaclav Kotesovec, Jul 28 2021: (Start)
a(n) = Bell(n+3) - 3*Bell(n+2) + 3*Bell(n+1) - Bell(n).
a(n) ~ n^3 * Bell(n) / LambertW(n)^3 * (1 - 3*LambertW(n)/n). (End)

A059099 Expansion of exp(exp(x)-1)/(2-exp(x)).

Original entry on oeis.org

1, 2, 7, 33, 198, 1453, 12669, 128320, 1482721, 19260421, 277913552, 4410640919, 76360030701, 1432144732762, 28926138244883, 625974400305541, 14449445989893990, 354384475357492593, 9202837263156670345, 252260867710562944224, 7278710072406887897461

Offset: 0

Vladeta Jovovic, Jan 02 2001

Row sums of A227343. - Peter Bala, Jul 11 2013

The sequence gives the number of barred preferential arrangements of an n-set having one bar, where one fixed section is a free section and elements which are to go into the other section are partitioned into unordered nonempty subsets. - Sithembele Nkonkobe, Jul 02 2015

			exp(exp(x)-1)/(2-exp(x)) = 1 + 2*x + 7/2*x^2 + 11/2*x^3 + 33/4*x^4 + 1453/120*x^5 + 4223/240*x^6 + 1604/63*x^7 + ...

Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, Combinatorial Identities for Vacillating Tableaux, arXiv:2308.14183 [math.CO], 2023. See pp. 12, 19, 29.
S. Nkonkobe and V. Murali, A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements, arXiv:1503.06172 [math.CO], 2015.

Row sums of A059098.

Cf. A001861, A000110, A005493, A049020, A227343.

s := series(exp(exp(x)-1)/(2-exp(x)), x, 60): for i from 0 to 50 do printf(`%d,`,i!*coeff(s,x,i)) od:

CoefficientList[Series[E^(E^x-1)/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 02 2015 *)

a(n) = Sum_{m=0..n} Sum_{i=0..n} Stirling2(n, i)*Product_{j=1..m} (i-j+1).
Stirling transform of A000522. - Vladeta Jovovic, May 10 2004
a(n) ~ n!*exp(1)/(2*(log(2))^(n+1)). - Vaclav Kotesovec, Jul 02 2015

More terms from James Sellers, Jan 03 2001

A102286 Total number of odd blocks in all partitions of n-set.

Original entry on oeis.org

1, 2, 7, 24, 96, 418, 1989, 10216, 56275, 330424, 2057672, 13532060, 93633021, 679473694, 5156626991, 40824399712, 336406367196, 2879570703510, 25557841113625, 234822774979908, 2230107923204443, 21861817965483016, 220940261740238140, 2299258336094622008

Offset: 1

Vladeta Jovovic, Feb 19 2005

a(n) is also the number of set partitions of {1,2,...,n+1} in which the element 1 is in an even size block. - Geoffrey Critzer, Apr 02 2013

			a(3)=7 because we have (123), (1)/23, 12/(3), 13/(2), (1)/(2)/(3); the odd blocks are shown between parentheses.

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

Cf. A000110, A000296, A005493.

G:=sinh(x)*exp(exp(x)-1): Gser:=series(G,x=0,30): seq(n!*coeff(Gser,x^n),n=1..25); # Emeric Deutsch
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
       add((p->(p+[0, `if`(i::odd, j, 0)*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[ (Cosh[x] - 1) + y Sinh[x]], {x, 0, nn}], y] /. y -> 1, x] (* Geoffrey Critzer, Aug 28 2012 *)
With[{nn=30},CoefficientList[Series[Sinh[x]Exp[Exp[x]-1],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 03 2021 *)

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

a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n,2*k+1) * Bell(n-2*k-1). - Ilya Gutkovskiy, Apr 10 2022

More terms from Emeric Deutsch, Mar 04 2005

A123158 Square array related to Bell numbers read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 5, 1, 52, 52, 37, 22, 6, 1, 203, 203, 151, 99, 31, 9, 1, 877, 877, 674, 471, 160, 61, 10, 1, 4140, 4140, 3263, 2386, 856, 385, 75, 14, 1, 21147, 21147, 17007, 12867, 4802, 2416, 520, 135, 15, 1

Offset: 0

Philippe Deléham, Oct 01 2006

			Square array, A(n, k), begins:
   1,   1,   1,    1,    1, ... (Row n=0: A000012);
   1,   2,   3,    5,    6, ... (Row n=1: A117142);
   2,   5,  10,   22,   31, ...;
   5,  15,  37,   99,  160, ...;
  15,  52, 151,  471,  856, ...;
  52, 203, 674, 2386, 4802, ...;
Antidiagonals, T(n, k), begin as:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   15,  15,  10,   5,   1;
   52,  52,  37,  22,   6,  1;
  203, 203, 151,  99,  31,  9,   1;
  877, 877, 674, 471, 160, 61,  10,  1;

G. C. Greubel, Antidiagonals n = 0..50, flattened

Columns include: A000110 (Bell numbers), A003128, A005493, A033452.

Rows include: A000012, A117142.

function A(n,k)
  if k lt 0 or n lt 0 then return 0;
  elif n eq 0 then return 1;
  elif (k mod 2) eq 1 then return A(n,k-1) + (1/2)*(k+1)*A(n-1,k+1);
  else return A(n,k-1) + A(n-1,k+1);
  end if;
end function;
T:= func< n,k | A(n-k,k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

A[0, _?NonNegative] = 1;
A[n_, k_]:= A[n, k]= If[n<0 || k<0, 0, If[OddQ[k], A[n, k-1] + (1/2)(k+1) A[n-1, k+1], A[n, k-1] + A[n-1, k+1]]];
Table[A[n-k, k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Feb 21 2020 *)

@CachedFunction
def A(n,k):
    if (k<0 or n<0): return 0
    elif (n==0): return 1
    elif (k%2==1): return A(n,k-1) +(1/2)*(k+1)*A(n-1,k+1)
    else: return A(n,k-1) +A(n-1,k+1)
def T(n,k): return A(n-k,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jul 18 2023

A(n, k) = 0 if n < 0, A(0, k) = 1 for k >= 0, A(n, k) = A(n, k-1) + (1/2)*(k+1)*A(n-1, k+1) if k is an odd number, A(n, k) = A(n, k-1) + A(n-1, k+1) if k is an even number (array).
A(n, 0) = A000110(n).
A(n, 1) = A000110(n+1).
A(n, 2) = A005493(n).
A(n, 3) = A033452(n).
A(n, 4) = A003128(n+2).
T(n, k) = A(n-k, k) (antidiagonals).

cp's OEIS Frontend

A138378 Number of embedded coalitions in an n-person game.

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A132963 Total number of distinct block sizes in all partitions of [n].

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A137650 Triangle read by rows, A008277 * A000012.

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A196834 Row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

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A190823 Number of permutations of 2 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 2.

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A362924 Triangle read by rows: T(n,m), n >= 1, 1 <= m <= n, is number of partitions of the set {1,2,...,n} that have at most one block contained in {1,...,m}.

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A011966 Third differences of Bell numbers.

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