A005493 -id:A005493 - cp's OEIS frontend

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

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

A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 5, 4, 1, 9, 19, 8, 1, 14, 55, 65, 16, 1, 20, 125, 285, 211, 32, 1, 27, 245, 910, 1351, 665, 64, 1, 35, 434, 2380, 5901, 6069, 2059, 128, 1, 44, 714, 5418, 20181, 35574, 26335, 6305, 256, 1, 54, 1110, 11130, 58107, 156660, 204205, 111645, 19171, 512

Offset: 1

Thomas Wieder, Dec 29 2006

Many well-known integer sequences arise from such a matrix product of combinatorial coefficients. In the present case we have as the first row A000079 = the powers of two = 2^n. As the second row we have A001047 = 3^n - 2^n. As the column sums we have 1,3,10,37,151,674,3263,17007,94828 we have A005493 = number of partitions of [n+1] with a distinguished block.

			Matrix begins:
1, 2, 4,  8, 16,  32,   64,  128,   256, ... A000079
0, 1, 5, 19, 65, 211,  665, 2059,  6305, ... A001047
0, 0, 1,  9, 55, 285, 1351, 6069, 26335, ... A016269
0, 0, 0,  1, 14, 125,  910, 5901, 35574, ... A025211
0, 0, 0,  0,  1,  20,  245, 2380, 20181, ...
0, 0, 0,  0,  0,   1,   27,  434,  5418, ...
0, 0, 0,  0,  0,   0,    1,   35,   714, ...
0, 0, 0,  0,  0,   0,    0,    1,    44, ...
0, 0, 0,  0,  0,   0,    0,    0,     1, ...
Triangle begins:
1;
1,  2;
1,  5,  4;
1,  9, 19,  8;
1, 14, 55, 65, 16;

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A039810, A039814, A126350, A054654, A126353.

T:= (n, k)-> add(binomial(n-1, i-1) *Stirling2(i, n+1-k), i=1..n):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Sep 29 2011

T[n_, k_] := Sum[Binomial[n-1, i-1]*StirlingS2[i, n+1-k], {i, 1, n}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)

(In Maple notation:) Matrix product B.A of matrix A[i,j]:=binomial(j-1,i-1) with i = 1 to p+1, j = 1 to p+1, p=8 and of matrix B[i,j]:=stirling2(j,i) with i from 1 to d, j from 1 to d, d=9.

T(n,k) = Sum_{i=1..n} C(n-1,i-1) * Stirling2(i, n+1-k). - Alois P. Heinz, Sep 29 2011

A232473 3-Fubini numbers.

Original entry on oeis.org

6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970

Offset: 3

N. J. A. Sloane, Nov 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 3..200
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.

r:=3; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(3);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 3], {n, 3, 22}] (* Jean-François Alcover, Mar 30 2016 *)

From Peter Bala, Dec 16 2020: (Start)
a(n+3) = Sum_{k = 0..n} (k+3)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n ).
a(n+3) = Sum_{k = 0..n} 3^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+3)! ).
E.g.f. with offset 0: 6*exp(3*z)/(2 - exp(z))^4 = 6 + 42*z + 342*z^2/2! + 3210*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

A232474 4-Fubini numbers.

Original entry on oeis.org

24, 216, 2184, 24696, 310344, 4304376, 65444424, 1083832056, 19437971784, 375544415736, 7779464328264, 172062025581816, 4047849158698824, 100946105980181496, 2660400563437957704, 73890563849015945976, 2157336929022064219464, 66059202473570840113656, 2116993226046938197020744

Offset: 4

N. J. A. Sloane, Nov 27 2013

Vincenzo Librandi, Table of n, a(n) for n = 4..200
Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A232472, A232473, A232474.

r:=4; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(4);

Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 4], {n, 4, 22}] (* Jean-François Alcover, Mar 30 2016 *)

From Peter Bala, Dec 16 2020: (Start)
a(n+4) = Sum_{k = 0..n} (k+4)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+4)^n ).
a(n+4) = Sum_{k = 0..n} 4^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+4)! ).
E.g.f. with offset 0: 24*exp(4*z)/(2 - exp(z))^5 = 24 + 216*z + 2184*z^2/2! + 24696*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020

A268814 Number of purely crossing partitions of [n].

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 5, 14, 62, 298, 1494, 8140, 47146, 289250, 1873304, 12756416, 91062073, 679616480, 5290206513, 42858740990, 360686972473, 3147670023632, 28439719809159, 265647698228954, 2561823514680235, 25475177517626196, 260922963832247729, 2749617210928715246

Offset: 0

Michel Marcus, Feb 14 2016

nonn

For the definition of a purely crossing partition refer to Dykema link (see PC(n) Definition 1.2 and Table 2).
From Gus Wiseman, Feb 23 2019: (Start)
For n >= 1, a set partition of {1,...,n} is purely crossing if it is topologically connected (A099947), has no successive elements in the same block (A000110(n - 1)), and the first and last vertices belong to different blocks (A005493(n - 2)). For example, the a(4) = 1, a(6) = 5, and a(7) = 14 purely crossing set partitions are:
  {{13}{24}}  {{135}{246}}    {{13}{246}{57}}
              {{13}{25}{46}}  {{13}{257}{46}}
              {{14}{25}{36}}  {{135}{26}{47}}
              {{14}{26}{35}}  {{135}{27}{46}}
              {{15}{24}{36}}  {{136}{24}{57}}
                              {{136}{25}{47}}
                              {{14}{257}{36}}
                              {{14}{26}{357}}
                              {{146}{25}{37}}
                              {{146}{27}{35}}
                              {{15}{246}{37}}
                              {{15}{247}{36}}
                              {{16}{24}{357}}
                              {{16}{247}{35}}
(End)

			G.f.: A(x) = 1 + x^4 + 5*x^6 + 14*x^7 + 62*x^8 + 298*x^9 + 1494*x^10 + 8140*x^11 + 47146*x^12 +...

Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.

Cf. A000108 (non-crossing partitions), A000110, A000699, A001263, A002662, A005493, A016098, A054726, A099947, A268815, A306417, A324011, A324166, A324172, A324173, A324324.

n = 30; F = x*Sum[BellB[k] x^k, {k, 0, n}] + O[x]^n; B = ComposeSeries[1/( InverseSeries[F, w]/w)-1, x/(1+x) + O[x]^n]; A = (B-x)/(1+x); Join[{1}, CoefficientList[A, x] // Rest] (* Jean-François Alcover, Feb 23 2016, adapted from K. J. Dykema's code *)
intvQ[set_]:=Or[set=={},Sort[set]==Range[Min@@set,Max@@set]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],And[!MatchQ[#,{_,{_,x_,y_,_},_}/;x+1==y],#=={}||And@@Not/@intvQ/@Union@@@Subsets[#,{1,Length[#]-1}],#=={}||Position[#,1][[1,1]]!=Position[#,n][[1,1]]]&]],{n,0,10}] (* Gus Wiseman, Feb 23 2019 *)

lista(nn) = {c = x/serreverse(x*serlaplace(exp(exp(x+x*O(x^nn)) -1))); b = subst(c, x, x/(1+x)+ O(x^nn)); vb = Vec(b-1); va = vector(#vb); va[1] = 0; va[2] = 0; for (k=3, #va, va[k] = vb[k] - va[k-1]; ); concat(1, va); }

{a(n) = my(A=1+x^3); for(i=1, n, A = sum(m=0, n, x^m/prod(k=1, m, (1+x)^2*A - k*x +x*O(x^n)) )/(1+x) ); polcoeff( A, n)}
for(n=0,35,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

{Stirling2(n, k) = n!*polcoeff(((exp(x+x*O(x^n)) - 1)^k)/k!, n)}
{Bell(n) = sum(k=0,n, Stirling2(n, k) )}
{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, Bell(m)*x^m/((1+x +x*O(x^n))^(2*m+1)*A^m)) ); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

G.f.: G(x) satisfies B(x) = x + (1 + x)*G(x) where B(x) is the g.f. of A268815 (see A(x) in Dykema link p. 7).
From Paul D. Hanna, Mar 07 2016: (Start)
O.g.f. A(x) satisfies:
(1) A(x) = Sum_{n>=0} A000110(n)*x^n / ((1+x)^(2*n+1) * A(x)^n), where A000110 are the Bell numbers.
(2) A(x) = 1/(1+x) * Sum_{n>=0} x^n / Product_{k=1..n} ((1+x)^2*A(x) - k*x).
(3) A(x) = 1/(1+x - x/((1+x)*A(x) - 1*x/(1+x - x/((1+x)*A(x) - 2*x/(1+x - x/((1+x)*A(x) - 3*x/(1+x - x/((1+x)*A(x) - 4*x/(1+x - x/((1+x)*A(x) -...)))))))))), a continued fraction. (End)

A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 20, 12, 4, 1, 75, 50, 20, 5, 1, 312, 225, 100, 30, 6, 1, 1421, 1092, 525, 175, 42, 7, 1, 7016, 5684, 2912, 1050, 280, 56, 8, 1, 37260, 31572, 17052, 6552, 1890, 420, 72, 9, 1, 211470, 186300, 105240, 42630, 13104, 3150, 600, 90, 10, 1

Offset: 1

Geoffrey Critzer, Dec 04 2010

The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.

T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.

			The set {1,2,3} has 5 partitions, {{1, 2, 3}}, {{2, 3}, {1}}, {{1, 3}, {2}}, {{1, 2}, {3}}, and {{2}, {3}, {1}}, and there are a total of 3 blocks of size 2, so T(3,2)=3.
Triangle begins:
    1;
    2,   1;
    6,   3,   1;
   20,  12,   4,  1;
   75,  50,  20,  5, 1;
  312, 225, 100, 30, 6, 1;
  ...

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

Cf. A000110, A056860, A052889, A070071, A105479, A105480, A105481, A105482, A285595.

b:= proc(n) option remember; `if`(n=0, [1, 0],
      add((p-> p+[0, p[1]*x^j])(b(n-j)*
      binomial(n-1, j-1)), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):
seq(T(n), n=1..12);  # Alois P. Heinz, Apr 24 2017

Table[Table[Length[Select[Level[SetPartitions[m],{2}],Length[#]==n&]],{n,1,m}],{m,1,10}]//Grid

E.g.f. for column k is x^k/k!*exp(exp(x)-1).

Sum_{k=1..n} k * T(n,k) = A070071(n). - Alois P. Heinz, Mar 03 2020

A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

Original entry on oeis.org

0, 0, 0, 1, 8, 47, 258, 1426, 8154, 48715, 305012, 2001719, 13754692, 98801976, 740584196, 5782218745, 46942426080, 395607218279, 3455493024350, 31236784338746, 291836182128670, 2814329123555051, 27980637362452980

Offset: 0

Emeric Deutsch, May 07 2010

nonn

Number of blocks not consisting of consecutive integers in all partitions of the set {1,2,...,n} (a singleton is considered a block of consecutive integers). Example: a(3)=1 because in 1-2-3, 1-23, 12-3, 13-2, and 123 only the block 13 does not consist of consecutive integers.

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A000110, A005493, A177254, A177255, A177256.

A177257:= func< n | n eq 0 select 0 else (&+[(Binomial(n,j)-(j+1))*Bell(j): j in [0..n-1]]) >;
[A177257(n): n in [0..30]]; // G. C. Greubel, May 12 2024

with(combinat): a:= proc(n) add((binomial(n, j)-j-1)*bell(j), j = 0 .. n-1) end proc: seq(a(n), n = 0 .. 22);

Table[Sum[(Binomial[n,j]-j-1)BellB[j],{j,0,n-1}],{n,0,30}] (* Harvey P. Dale, Oct 15 2015 *)

def A177257(n): return sum((binomial(n,j) -(j+1))*bell_number(j) for j in range(n))
[A177257(n) for n in range(31)] # G. C. Greubel, May 12 2024

a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*Bell(j), where Bell(n) = A000110(n) are the Bell numbers.
a(n) = Sum_{k=0..floor(n/2)} k*A177256(n,k).
a(n) = A005493(n-1) - A177255(n).

A239145 Number T(n,k) of self-inverse permutations p on [n] where the minimal transposition distance equals k (k=0 for the identity permutation); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 13, 8, 3, 1, 0, 1, 39, 22, 10, 3, 1, 0, 1, 120, 65, 32, 10, 3, 1, 0, 1, 401, 208, 103, 37, 10, 3, 1, 0, 1, 1385, 703, 344, 136, 37, 10, 3, 1, 0, 1, 5069, 2517, 1206, 501, 151, 37, 10, 3, 1, 0, 1, 19170, 9390, 4421, 1890, 622, 151, 37, 10, 3, 1, 0

Offset: 0

Joerg Arndt and Alois P. Heinz, Mar 11 2014

Columns k=0 and k=1 respectively give A000012 and A000085(n)-A170941(n).
Row sums give A000085.
Diagonal T(2n,n) gives A005493(n-1) for n>0.
Reversed rows converge to A005493.

			T(4,0) = 1: 1234.
T(4,1) = 5: 1243, 1324, 2134, 2143, 4321.
T(4,2) = 3: 1432, 3214, 3412.
T(4,3) = 1: 4231.
Triangle T(n,k) begins:
00:   1;
01:   1,    0;
02:   1,    1,    0;
03:   1,    2,    1,    0;
04:   1,    5,    3,    1,   0;
05:   1,   13,    8,    3,   1,   0;
06:   1,   39,   22,   10,   3,   1,  0;
07:   1,  120,   65,   32,  10,   3,  1,  0;
08:   1,  401,  208,  103,  37,  10,  3,  1, 0;
09:   1, 1385,  703,  344, 136,  37, 10,  3, 1, 0;
10:   1, 5069, 2517, 1206, 501, 151, 37, 10, 3, 1, 0;

Joerg Arndt and Alois P. Heinz, Rows n = 0..30, flattened

b:= proc(n, k, s) option remember; `if`(n=0, 1, `if`(n in s,
      b(n-1, k, s minus {n}), b(n-1, k, s) +add(`if`(i in s, 0,
      b(n-1, k, s union {i})), i=1..n-k-1)))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, k-1, {})-b(n, k, {})):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, k_, s_List] := b[n, k, s] = If[n == 0, 1, If[MemberQ[s, n], b[n-1, k, s ~Complement~ {n}], b[n-1, k, s] + Sum[If[MemberQ[s, i], 0, b[n-1, k, s ~Union~ {i}]], {i, 1, n - k - 1}]]] ; T[n_, k_] := If[k == 0, 1, b[n, k-1, {}] - b[n, k, {}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Maple *)

T(n,k) = A239144(n,k-1) - A239144(n,k) for k>0, T(n,0) = 1.

A268815 Number of purely crossing + partitions of [n].

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 5, 19, 76, 360, 1792, 9634, 55286, 336396, 2162554, 14629720, 103818489, 770678553, 5969822993, 48148947503, 403545713463, 3508356996105, 31587389832791, 294087418038113, 2827471212909189, 28037001032306431, 286398141349873925, 3010540174760962975

Offset: 0

Michel Marcus, Feb 14 2016

nonn

For the definition of these special purely crossing partitions refer to Dykema link (see PC+(n) Definition 2.1 and Table 2).
From Gus Wiseman, Feb 23 2019: (Start)
a(n) is the number of topologically connected (A099947) set partitions of {1,...,n} with no successive elements in the same block. For example, the a(4) = 1 through a(7) = 19 set partitions are:
  {{13}{24}}  {{135}{24}}  {{135}{246}}    {{1357}{246}}
                           {{13}{25}{46}}  {{13}{246}{57}}
                           {{14}{25}{36}}  {{13}{257}{46}}
                           {{14}{26}{35}}  {{135}{26}{47}}
                           {{15}{24}{36}}  {{135}{27}{46}}
                                           {{136}{24}{57}}
                                           {{136}{25}{47}}
                                           {{137}{25}{46}}
                                           {{14}{257}{36}}
                                           {{14}{26}{357}}
                                           {{146}{25}{37}}
                                           {{146}{27}{35}}
                                           {{147}{25}{36}}
                                           {{147}{26}{35}}
                                           {{15}{246}{37}}
                                           {{15}{247}{36}}
                                           {{157}{24}{36}}
                                           {{16}{24}{357}}
                                           {{16}{247}{35}}
(End)

			G.f.: A(x) = 1 + x + x^4 + x^5 + 5*x^6 + 19*x^7 + 76*x^8 + 360*x^9 + 1792*x^10 +...

Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016.

Cf. A000108, A000110, A005493, A016098, A099947, A268814, A306417, A324011, A324166, A324173, A324324, A324327.

n = 30; F = x*Sum[BellB[k] x^k, {k, 0, n}] + O[x]^n; B = ComposeSeries[1/( InverseSeries[F, w] /w)-1, x/(1+x) + O[x]^n]; CoefficientList[B, x] // Rest (* Jean-François Alcover, Feb 16 2016, adapted from K. J. Dykema's code *)
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
intvQ[set_]:=Or[set=={},Sort[set]==Range[Min@@set,Max@@set]];
Table[Length[Select[sps[Range[n]],And[!MatchQ[#,{_,{_,x_,y_,_},_}/;x+1==y],#=={}||And@@Not/@intvQ/@Union@@@Subsets[#,{1,Length[#]-1}]]&]],{n,0,10}] (* Gus Wiseman, Feb 23 2019 *)

lista(nn) = {c = x/serreverse(x*serlaplace(exp(exp(x+x*O(x^nn)) -1))); b = subst(c, x, x/(1+x) + O(x^nn)); Vec(b);}

{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/prod(k=1, m, (1+x)*A - k*x +x*O(x^n)) )); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

{Stirling2(n, k) = n!*polcoeff(((exp(x+x*O(x^n)) - 1)^k)/k!, n)}
{Bell(n) = sum(k=0,n, Stirling2(n, k) )}
{a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, Bell(m)*x^m/((1+x)*A +x*O(x^n))^m) ); polcoeff(A, n)}
for(n=0,25,print1(a(n),", ")) \\ Paul D. Hanna, Mar 07 2016

G.f.: G(x) satisfies C(x) = G(x/1-x) where C(x) is the g.f. of A099947 (see B(x) in Dykema link p. 7).
From Paul D. Hanna, Mar 07 2016: (Start)
O.g.f. A(x) satisfies
(1) A(x) = Sum_{n>=0} A000110(n)*x^n/((1+x)^n*A(x)^n), where A000110 are the Bell numbers.
(2) A(x) = Sum_{n>=0} x^n / Product_{k=1..n} ((1+x)*A(x) - k*x).
(3) A(x) = 1/(1 - x/((1+x)*A(x) - 1*x/(1 - x/((1+x)*A(x) - 2*x/(1 - x/((1+x)*A(x) - 3*x/(1 - x/((1+x)*A(x) - 4*x/(1 - x/((1+x)*A(x) - ... )))))))))), a continued fraction. (End)

A276974 Number T(n,k) of permutations of [n] where the minimal distance between elements of the same cycle equals k (k=n for the identity permutation in S_n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 1, 1, 0, 19, 3, 1, 1, 0, 103, 12, 3, 1, 1, 0, 651, 54, 10, 3, 1, 1, 0, 4702, 281, 42, 10, 3, 1, 1, 0, 38413, 1652, 203, 37, 10, 3, 1, 1, 0, 350559, 11017, 1086, 166, 37, 10, 3, 1, 1, 0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1, 0, 39196758, 669948, 44265, 4900, 726, 151, 37, 10, 3, 1, 1

Offset: 0

Alois P. Heinz, Sep 23 2016

			T(3,1) = 4: (1,2,3), (1,3,2), (1)(2,3), (1,2)(3).
T(3,2) = 1: (1,3)(2).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,       1;
  0,       1,     1;
  0,       4,     1,    1;
  0,      19,     3,    1,   1;
  0,     103,    12,    3,   1,   1;
  0,     651,    54,   10,   3,   1,  1;
  0,    4702,   281,   42,  10,   3,  1,  1;
  0,   38413,  1652,  203,  37,  10,  3,  1, 1;
  0,  350559, 11017, 1086, 166,  37, 10,  3, 1, 1;
  0, 3539511, 81665, 6564, 857, 151, 37, 10, 3, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..12, flattened
Per Alexandersson et al., d-regular partitions and permutations, MathOverflow, 2014

Columns k=0-1 give: A000007, A276975.
Row sums give A000142.
T(2n,n) = A138378(n) = A005493(n-1) for n>0.
Cf. A239145, A263757, A277031.

cp's OEIS Frontend

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

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A126351 Triangle read by rows: matrix product of the Stirling numbers of the second kind with the binomial coefficients.

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A232473 3-Fubini numbers.

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A232474 4-Fubini numbers.

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A268814 Number of purely crossing partitions of [n].

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A175757 Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.

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A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

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