A264428 -id:A264428 - cp's OEIS frontend

A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

1, 8, 1, 120, 24, 1, 2640, 672, 48, 1, 76560, 22800, 2160, 80, 1, 2756160, 920160, 104880, 5280, 120, 1, 118514880, 43243200, 5639760, 347760, 10920, 168, 1, 5925744000, 2323918080, 336510720, 24071040, 937440, 20160, 224, 1

Offset: 1

Wolfdieter Lang Sep 14 2007

Previous name was: Triangle of numbers related to triangle A132057; generalization of Stirling numbers of second kind A008277, Lah-numbers A008297, ...
a(n,m) enumerates unordered n-vertex m-forests composed of m plane increasing 8-ary trees. See the F. Bergeron et al. reference, especially Table 1, first row, for the e.g.f. for m=1.
a(n,m) := S2(8; n,m) is the eighth triangle of numbers in the sequence S2(k;n,m), k=1..7: A008277 (unsigned Stirling 2nd kind), A008297 (unsigned Lah), A035342, A035469, A049029, A049385, A092082, respectively. a(n,1)=A045754(n), n>=1.

			{1}; {8,1}; {120,24,1}; {2640,672,48,1}; ...

F. Bergeron, Ph. Flajolet and B. Salvy, Varieties of Increasing Trees, Lecture Notes in Computer Science vol. 581, ed. J.-C. Raoult, Springer 1992, pp. 24-48.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
W. Lang, First 10 rows.

Cf. A132060 (row sums), A132061 (alternating row sums).

Cf. A092082 S2(7) triangle.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(7*k+1, k=0..n), 8); # Peter Luschny, Jan 27 2016

a[n_, m_] := a[n, m] = ((m*a[n-1, m-1]*(m-1)! + (m+7*n-7)*a[n-1, m]*m!)*n!)/(n*m!*(n-1)!);
a[n_, m_] /; n < m = 0; a[_, 0] = 0; a[1, 1] = 1;
Flatten[Table[a[n, m], {n, 1, 8}, {m, 1, n}]][[1 ;; 36]]
(* Jean-François Alcover, Jun 17 2011 *)
rows = 8;
a[n_, m_] := BellY[n, m, Table[Product[7k+1, {k, 0, j}], {j, 0, rows}]];
Table[a[n, m], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

a(n, m) = n!*A132057(n, m)/(m!*7^(n-m)); a(n+1, m) = (7*n+m)*a(n, m)+ a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. of m-th column: ((-1+(1-7*x)^(-1/7))^m)/m!.

a(n, m) = sum(|A051186(n, j)|*S2(j, m), j=m..n) (matrix product), with S2(j, m):= (j, m) (Stirling2 triangle). Priv. comm. with W. Lang by E. Neuwirth, Feb 15 2001; see also the 2001 Neuwirth reference. See the general comment on products of Jabotinsky matrices given under A035342.

New name from Peter Luschny, Jan 27 2016

A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

Original entry on oeis.org

1, 9, 1, 153, 27, 1, 3825, 855, 54, 1, 126225, 32895, 2745, 90, 1, 5175225, 1507815, 150930, 6705, 135, 1, 253586025, 80565975, 9205245, 499590, 13860, 189, 1, 14454403425, 4926412575, 623675430, 39180645, 1345050, 25578, 252, 1

Offset: 1

Udita Katugampola, Mar 23 2013

Also the Bell transform of A045755(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 29 2016

			1;
9,1;
153,27,1;
3825,855,54,1;
126225,32895,2745,90,1;
5175225,1507815,150930,6705,135,1;
253586025,80565975,9205245,499590,13860,189,1;
14454403425,4926412575,623675430,39180645,1345050,25578,252,1;

Cf. A008277, A019538, A035342, A035469, A049029, A049385, A092082, A132056, A223512-A223522, A223168-A223172, A223523-A223532.

b[0]:=g(x):
for j from 1 to 10 do
b[j]:=simplify(x^9*diff(b[j-1],x$1);
end do;
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> mul(8*k+1, k=0..n), 10); # Peter Luschny, Jan 29 2016

rows = 8;
t = Table[Product[8k+1, {k, 0, n}], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 16, 15, 6, 1, 125, 110, 45, 10, 1, 1296, 1080, 435, 105, 15, 1, 16807, 13377, 5250, 1295, 210, 21, 1, 262144, 200704, 76608, 18865, 3220, 378, 28, 1, 4782969, 3542940, 1316574, 320544, 55755, 7056, 630, 36, 1, 100000000, 72000000, 26100000, 6258000, 1092105, 143325, 14070, 990, 45, 1

Offset: 1

Washington Bomfim, Apr 14 2005; revised May 19 2005

Row sums equal A001858 (number of forests of labeled trees with n nodes).
Also the Bell transform of A000272(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
The permutohedron (convex hull of permutations on 1,...,n in R^n) has Ehrhart polynomial Sum_{k=0..n-1} T(n,n-k) t^k. - Matthieu Josuat-Vergès, Mar 31 2018

			T(3, 2) = 3 because there are 3 such forests with 3 nodes and 2 trees.
Triangle begins:
      1;
      1,     1;
      3,     3,    1;
     16,    15,    6,    1;
    125,   110,   45,   10,   1;
   1296,  1080,  435,  105,  15,  1;
  16807, 13377, 5250, 1295, 210, 21, 1;

B. Bollobas, Graph Theory - An Introductory Course (Springer-Verlag, New York, 1979)

Alois P. Heinz, Rows n = 1..141, flattened
E. Britikov, Asymptotic number of forests from unrooted trees, Mat. Zametki (1988).
Mathoverflow, Is there a formula for the number of labeled forests with k components on n vertices?

Cf. A033185, A106240.
Rows reflected give A138464. - Alois P. Heinz, Sep 10 2008
Columns k=1-10 give: A000272, A083483, A239910, A240681, A240682, A240683, A240684, A240685, A240686, A240687.
T(2n,n) gives A302112.

Flat(List([1..11],n->List([1..n],m->(1/Factorial(m)*Sum([0..m],j->(-1/2)^j*Binomial(m,j)*Binomial(n-1,m+j-1)*n^(n-m-j)*Factorial(m+j)))))); # Muniru A Asiru, Apr 01 2018

T:= proc(n,m) option remember;
      if n<0 then 0
    elif n=m then 1
    elif m<1 or m>n then 0
    else add(binomial(n-1,j-1)*j^(j-2)*T(n-j,m-1), j=1..n-m+1)
      fi
    end:
seq(seq(T(n, m), m=1..n), n=1..12); # Alois P. Heinz, Sep 10 2008
# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> (n+1)^(n-1), 9); # Peter Luschny, Jan 27 2016

f[list_]:=Select[list,#>0&];Flatten[Map[f, Transpose[Table[t = Sum[n^(n - 2) x^n/n!, {n, 1, 20}];Drop[Range[0, 8]! CoefficientList[Series[t^k/k!, {x, 0, 8}], x],1], {k, 1, 8}]]]] (* Geoffrey Critzer, Nov 22 2011 *)
T[n_, m_] := Sum[(-1/2)^j*Binomial[m, j]*Binomial[n-1, m+j-1]*n^(n-m-j)*(m + j)!, {j, 0, m}]/m!; Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Max Alekseyev *)
rows = 10;
t = Table[(n+1)^(n-1), {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

{ T(n,m) = sum(j=0,m, (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)! )/m! } /* Max Alekseyev, Oct 08 2014 */

T(n,m) = Sum_{k=1..n-m+1} binomial(n-1,k-1)*k^(k-2)*T(n-k,m-1), T(n,0) = 0 if n > 0, T(0,0) = 1. - Vladeta Jovovic and Washington Bomfim
The value of T(n, m) can be calculated by the formula in Bollobas, pp. 172, exercise 44. Also T(n, m) = sum N/D over the partitions of n, 1*K(1) + 2*K(2) + ... + n*K(n), with exactly m parts, where N = n! * Product_{i = 1..n} i^( (i-2) * K(i) ) and D = Product_{i = 1..n} ( K(i)! * (i!)^K(i) ).
From Peter Bala, Aug 14 2012: (Start)
E.g.f.: A(x,t) := exp(t*F(x)) = 1 + t*x + (t + t^2)*x^2/2! + (3*t + 3*t^2 + t^3)*x^3/3! + ..., where F(x) = sum {n >= 1} n^(n-2)*x^n/n! is the e.g.f. for labeled trees (see A000272). The row polynomials R(n,t) are thus a sequence of binomial type polynomials.
Differentiating A(x,t) w.r.t. x yields A'(x,t) = t*A(x,t)*F'(x) leading to the recurrence equation for the row polynomials R(n,t) = t*sum {k = 0..n-1} (k+1)^(k-1)*binomial(n-1,k)*R(n-k-1,t) with R(0,t) = 1 and R(1,t) = t: the above recurrence for the table entries follows from this.
(End)
T(n,m) = (1/m!) * Sum_{j=0..m} (-1/2)^j * binomial(m,j) * binomial(n-1,m+j-1) * n^(n-m-j)* (m+j)!. Due to A. Renyi. - Max Alekseyev, Oct 08 2014
T(n,m) = (n!/m!)*Sum_{k_1+...+k_m=n, k_i>=1} Product_{j=1..m} k_j^(k_j-2)/k_j!. See Britikov reference. - Roland Vincze, Apr 18 2020

A122848 Exponential Riordan array (1, x(1+x/2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 15, 10, 1, 0, 0, 0, 15, 45, 15, 1, 0, 0, 0, 0, 105, 105, 21, 1, 0, 0, 0, 0, 105, 420, 210, 28, 1, 0, 0, 0, 0, 0, 945, 1260, 378, 36, 1, 0, 0, 0, 0, 0, 945, 4725, 3150, 630, 45, 1, 0, 0, 0, 0, 0, 0, 10395, 17325, 6930, 990, 55, 1, 0, 0

Offset: 0

Paul Barry, Sep 14 2006

Entries are Bessel polynomial coefficients. Row sums are A000085. Diagonal sums are A122849. Inverse is A122850. Product of A007318 and A122848 gives A100862.
T(n,k) is the number of self-inverse permutations of {1,2,...,n} having exactly k cycles. - Geoffrey Critzer, May 08 2012
Bessel numbers of the second kind. For relations to the Hermite polynomials and the Catalan (A033184 and A009766) and Fibonacci (A011973, A098925, and A092865) matrices, see Yang and Qiao. - Tom Copeland, Dec 18 2013.
Also the inverse Bell transform of the double factorial of odd numbers Product_{k= 0..n-1} (2*k+1) (A001147). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

			Triangle begins:
    1
    0    1
    0    1    1
    0    0    3    1
    0    0    3    6    1
    0    0    0   15   10    1
    0    0    0   15   45   15    1
    0    0    0    0  105  105   21    1
    0    0    0    0  105  420  210   28    1
    0    0    0    0    0  945 1260  378   36    1
From _Gus Wiseman_, Jan 12 2021: (Start)
As noted above, a(n) is the number of set partitions of {1..n} into k singletons or pairs. This is also the number of set partitions of subsets of {1..n} into n - k pairs. In the first case, row n = 5 counts the following set partitions:
  {{1},{2,3},{4,5}}  {{1},{2},{3},{4,5}}  {{1},{2},{3},{4},{5}}
  {{1,2},{3},{4,5}}  {{1},{2},{3,4},{5}}
  {{1,2},{3,4},{5}}  {{1},{2,3},{4},{5}}
  {{1,2},{3,5},{4}}  {{1,2},{3},{4},{5}}
  {{1},{2,4},{3,5}}  {{1},{2},{3,5},{4}}
  {{1},{2,5},{3,4}}  {{1},{2,4},{3},{5}}
  {{1,3},{2},{4,5}}  {{1},{2,5},{3},{4}}
  {{1,3},{2,4},{5}}  {{1,3},{2},{4},{5}}
  {{1,3},{2,5},{4}}  {{1,4},{2},{3},{5}}
  {{1,4},{2},{3,5}}  {{1,5},{2},{3},{4}}
  {{1,4},{2,3},{5}}
  {{1,4},{2,5},{3}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}
In the second case, we have:
  {{1,2},{3,4}}  {{1,2}}  {}
  {{1,2},{3,5}}  {{1,3}}
  {{1,2},{4,5}}  {{1,4}}
  {{1,3},{2,4}}  {{1,5}}
  {{1,3},{2,5}}  {{2,3}}
  {{1,3},{4,5}}  {{2,4}}
  {{1,4},{2,3}}  {{2,5}}
  {{1,4},{2,5}}  {{3,4}}
  {{1,4},{3,5}}  {{3,5}}
  {{1,5},{2,3}}  {{4,5}}
  {{1,5},{2,4}}
  {{1,5},{3,4}}
  {{2,3},{4,5}}
  {{2,4},{3,5}}
  {{2,5},{3,4}}
(End)

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
H. Han and S. Seo, Combinatorial proofs of inverse relations and log-concavity for Bessel numbers, Eur. J. Combinat. 29 (7) (2008) 1544-1554. [From _R. J. Mathar_, Mar 20 2009]
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 18.
S. Yang and Z. Qiao, The Bessel Numbers and Bessel Matrices, Journal of Mathematical Research & Exposition, July, 2011, Vol. 31, No. 4, pp. 627-636. [From _Tom Copeland_, Dec 18 2013]

Cf. A001497, A008275, A039755, A096713, A104556, A113278, A130757.
Row sums are A000085.
Column sums are A001515.
Same as A049403 but with a first column k = 0.
The same set partitions counted by number of pairs are A100861.
Reversing rows gives A111924 (without column k = 0).
A047884 counts standard Young tableaux by size and greatest row length.
A238123 counts standard Young tableaux by size and least row length.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
Cf. A000110, A000258, A320732, A321729, A339741, A339887.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n<2,1,0), 9); # Peter Luschny, Jan 27 2016

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); Table[ t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten
(* Second program: *)
rows = 12;
t = Join[{1, 1}, Table[0, rows]];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 23 2018,after Peter Luschny *)
sbs[{}]:={{}};sbs[set:{i_,_}]:=Join@@Function[s,(Prepend[#1,s]&)/@sbs[Complement[set,s]]]/@Cases[Subsets[set],{i}|{i,_}];
Table[Length[Select[sbs[Range[n]],Length[#]==k&]],{n,0,6},{k,0,n}] (* Gus Wiseman, Jan 12 2021 *)

{T(n,k)=if(2*kn, 0, n!/(2*k-n)!/(n-k)!*2^(k-n))} /* Michael Somos, Oct 03 2006 */

# uses[inverse_bell_transform from A265605]
multifact_2_1 = lambda n: prod(2*k + 1 for k in (0..n-1))
inverse_bell_matrix(multifact_2_1, 9) # Peter Luschny, Dec 31 2015

Number triangle T(n,k) = k!*C(n,k)/((2k-n)!*2^(n-k)).
T(n,k) = A001498(k,n-k). - Michael Somos, Oct 03 2006
E.g.f.: exp(y(x+x^2/2)). - Geoffrey Critzer, May 08 2012
Triangle equals the matrix product A008275*A039755. Equivalently, the n-th row polynomial R(n,x) is given by the Type B Dobinski formula R(n,x) = exp(-x/2)*Sum_{k>=0} P(n,2*k+1)*(x/2)^k/k!, where P(n,x) = x*(x-1)*...*(x-n+1) denotes the falling factorial polynomial. Cf. A113278. - Peter Bala, Jun 23 2014
From Daniel Checa, Aug 28 2022: (Start)
E.g.f. for the m-th column: (x^2/2+x)^m/m!.
T(n,k) = T(n-1,k-1) + (n-1)*T(n-2,k-1) for n>1 and k=1..n, T(0,0) = 1. (End)

A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

Original entry on oeis.org

1, -2, 1, 8, -6, 1, -48, 44, -12, 1, 384, -400, 140, -20, 1, -3840, 4384, -1800, 340, -30, 1, 46080, -56448, 25984, -5880, 700, -42, 1, -645120, 836352, -420224, 108304, -15680, 1288, -56, 1, 10321920, -14026752, 7559936, -2153088, 359184, -36288, 2184, -72, 1

Offset: 1

Wouter Meeussen

T(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.
Exponential Riordan array [1/(1+2x),log(1+2x)/2]. The unsigned triangle is [1/(1-2x),log(1/sqrt(1-2x))]. - Paul Barry_, Apr 29 2009
The n-th row is related to the expansion of z^(-2n)*(z^3 d/dz)^n in polynomials of the Euler operator D=(z d/dz). E.g., z^(-6)(z^3 d/dz)^3 = D^3 + 6 D^2 + 8 D. See Copeland link for relations to Bell / Exponential / Touchard polynomial operators. - Tom Copeland, Nov 14 2013
A refinement of this array is given by A231846. - Tom Copeland, Nov 15 2013
Also the Bell transform of the double factorial of even numbers A000165 except that the values are unsigned and in addition a first column (1,0,0 ...) is added on the left side of the triangle. For the Bell transform of the double factorial of odd numbers A001147 see A132062. For the definition of the Bell transform see A264428. - Peter Luschny, Dec 20 2015
The signed triangle is also the inverse Bell transform of A000079 (see Luschny link). - John Keith, Nov 24 2020

			Triangle starts:
  {1},
  {2,1},
  {8,6,1},
  {48,44,12,1},
  ...
From _Paul Barry_, Apr 29 2009: (Start)
The unsigned triangle [1/(1-2x),log(1/sqrt(1-2x))] has production matrix:
  2, 1,
  4, 4, 1,
  8, 12, 6, 1,
  16, 32, 24, 8, 1,
  32, 80, 80, 40, 10, 1,
  64, 192, 240, 160, 60, 12, 1
which is A007318^{2} beheaded. (End)

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales. Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Tom Copeland, Addendum to Mathemagical Forests.
P. Feijão, F. V. Martinez, and A. Thévenin, On the distribution of cycles and paths in multichromosomal breakpoint graphs and the expected value of rearrangement distance, BMC Bioinformatics 16:Suppl19 (2015), S1. doi:10.1186/1471-2105-16-S19-S1
Lisa Glaser, Causal set actions in various dimensions, J. Phys.: Conf. Ser. 306 (2011), 012041.
Wolfdieter Lang, First 9 rows and comment.
Peter Luschny, The Bell transform
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1).
Cf. A051141, A051142.
Cf. A000165, A132062, A264428.
Cf. A038207.

Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]

# uses[bell_transform from A264428]
# Unsigned values and an additional first column (1,0,0,...).
def A039683_unsigned_row(n):
    a = sloane.A000165
    dblfact = a.list(n)
    return bell_transform(n, dblfact)
[A039683_unsigned_row(n) for n in (0..9)] # Peter Luschny, Dec 20 2015

T(n, m) = T(n-1, m-1) - 2*(n-1)*T(n-1, m), n >= m >= 1; T(n, m) := 0, n

E.g.f. for m-th column of signed triangle: (((log(1+2*x))/2)^m)/m!.

E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} Stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic, Feb 11 2003

T(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

The production matrix below is A038207 with the first row removed. With the initial index n = 0, the associated differential raising operator is R = e^(2D)*x = (2+x)*e^(2D) with D = d/dx, i.e., R p_n(x) = p_(n+1)(x) where p_n(x) is the n-th unsigned row polynomial and p_0(x) = 1, so p_(n+1)(x) = (2+x) * p_n(2+x). - Tom Copeland, Oct 11 2016

Additional comments from Wolfdieter Lang

Title revised by Tom Copeland, Dec 21 2013

A059297 Triangle of idempotent numbers binomial(n,k)*k^(n-k), version 1.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 6, 1, 0, 4, 24, 12, 1, 0, 5, 80, 90, 20, 1, 0, 6, 240, 540, 240, 30, 1, 0, 7, 672, 2835, 2240, 525, 42, 1, 0, 8, 1792, 13608, 17920, 7000, 1008, 56, 1, 0, 9, 4608, 61236, 129024, 78750, 18144, 1764, 72, 1, 0, 10, 11520, 262440

Offset: 0

N. J. A. Sloane, Jan 25 2001

T(n,k) = C(n,k)*k^(n-k) is the number of functions f from domain [n] to codomain [n+1] such that f(x)=n+1 for exactly k elements x of [n] and f(f(x))=n+1 for the remaining n-k elements x of [n]. Subsequently, row sums of T(n,k) provide the number of functions f:[n]->[n+1] such that either f(x)=n+1 or f(f(x))=n+1 for every x in [n]. We note that there are C(n,k) ways to choose the k elements mapped to n+1 and there are k^(n-k) ways to map n-k elements to a set of k elements. - Dennis P. Walsh, Sep 05 2012
Conjecture: the matrix inverse is A137452. - R. J. Mathar, Mar 12 2013
The above conjecture is correct. This triangle is the exponential Riordan array [1, x*exp(x)]. Thus the  inverse array is the exponential Riordan array [ 1, W(x)], which equals A137452. - Peter Bala, Apr 08 2013

			Triangle begins:
1;
0,  1;
0,  2,   1;
0,  3,   6,    1;
0,  4,  24,   12,    1;
0,  5,  80,   90,   20,   1;
0,  6, 240,  540,  240,  30,  1;
0,  7, 672, 2835, 2240, 525, 42,  1;
Row 4. Expansion of x^4 in terms of Abel polynomials:
x^4 = -4*x+24*x*(x+2)-12*x*(x+3)^2+x*(x+4)^3.
O.g.f. for column 2: A(-2,1/x) = x^2/(1-2*x)^3 = x^2+6*x^3+24*x^4+80*x^5+....

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43 and p. 135, [3i'].

Alois P. Heinz, Rows n = 0..140, flattened
Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics, arXiv:quant-ph/0401126, 2004.
Emanuele Munarini, Combinatorial identities involving the central coefficients of a Sheffer matrix, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
Bruce E. Sagan, A note on Abel polynomials and rooted labeled forests. Discrete Mathematics 44(3): 293-298 (1983).
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 96, [_Tom Copeland_, Dec 20 2018].
Eric Weisstein's World of Mathematics, Abel Polynomial
Eric Weisstein's World of Mathematics, Idempotent Number
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

There are 4 versions: A059297, A059298, A059299, A059300.
Diagonals give A001788, A036216, A040075, A050982, A002378, 3*A002417, etc.
Row sums are A000248.
Cf. A061356, A202017, A137452 (inverse array), A264428.

/* As triangle */ [[Binomial(n,k)*k^(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Aug 22 2015

T:= (n, k)-> binomial(n, k) *k^(n-k):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 05 2012

nn=10;f[list_]:=Select[list,#>0&];Prepend[Map[Prepend[#,0]&,Rest[Map[f,Range[0,nn]!CoefficientList[Series[Exp[y x Exp[x]],{x,0,nn}],{x,y}]]]],{1}]//Grid  (* Geoffrey Critzer, Feb 09 2013 *)
t[n_, k_] := Binomial[n, k]*k^(n - k); Prepend[Flatten@Table[t[n, k], {n, 10}, {k, 0, n}], 1] (* Arkadiusz Wesolowski, Mar 23 2013 *)

# uses[bell_transform from A264428]
def A059297_row(n):
    nat = [k for k in (1..n)]
    return bell_transform(n, nat)
[A059297_row(n)  for n in range(8)] # Peter Luschny, Dec 20 2015

E.g.f.: exp(x*y*exp(y)). - Vladeta Jovovic, Nov 18 2003
Up to signs, this is the triangle of connection constants expressing the monomials x^n as a linear combination of the Abel polynomials A(k,x) := x*(x+k)^(k-1), 0 <= k <= n. O.g.f. for the k-th column: A(-k,1/x) = x^k/(1-k*x)^(k+1). Cf. A061356. Examples are given below. - Peter Bala, Oct 09 2011
The o.g.f.'s for the diagonals of this triangle are the rational functions occurring in the expansion of the compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... For example, the o.g.f. for second subdiagonal is (3*t+9*t^2)/(1-t)^5 = 3*t + 24*t^2 + 90*t^3 + 240*t^4 + .... See the Bala link. The coefficients of the numerator polynomials are listed in A202017. - Peter Bala, Dec 08 2011
Recurrence equation: T(n+1,k+1) = Sum_{j=0..n-k} (j+1)*binomial(n,j)*T(n-j,k). - Peter Bala, Jan 13 2015
The Bell transform of [1,2,3,...]. See A264428 for the Bell transform. - Peter Luschny, Dec 20 2015

A143395 Triangle read by rows: T(n,k) = number of forests of k labeled rooted trees of height at most 1, with n labels, where any root may contain >= 1 labels, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 7, 9, 1, 0, 15, 55, 18, 1, 0, 31, 285, 205, 30, 1, 0, 63, 1351, 1890, 545, 45, 1, 0, 127, 6069, 15421, 7770, 1190, 63, 1, 0, 255, 26335, 116298, 95781, 24150, 2282, 84, 1, 0, 511, 111645, 830845, 1071630, 416451, 62370, 3990, 108, 1

Offset: 0

Alois P. Heinz, Aug 12 2008

This is the Sheffer triangle (1,exp(x)*(exp(x)-1)) (Jobotinsky type). See the e.g.f. given by V. Jovovic below, and the W. Lang link under A006232 (second part) for general Sheffer remarks and the conversion to the umbral notation of S. Roman's book. - Wolfdieter Lang, Oct 08 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) counts the ways a set of size n can be partitioned into k nonempty blocks and then a nonempty subset chosen from each block. An example is given below.
This triangle is the particular case a = 1, b = 1, c = 0 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. A008277 is the case a = 1, b = 0, c = 0.
Define a polynomial sequence x_(n) by putting x_(0) = 1, x_(1) = x and for n >= 2 setting x_(n) = x*(x - (n+1))*(x - (n+2))*...*(x - (2*n-1)), that is, x_(n) = (-1)^(n+1)*n!*(x/(2*n - x))*binomial(2*n - x,n) for n >= 0. Then this table is the triangle of connection constants for expressing the monomial polynomials x^n in terms of the basis polynomials x_(k), that is, x^n = sum {k = 0..n} T(n,k)*x_(k), n = 0,1,2,.... Examples are given below.
Matrix factorization: Let M be the infinite lower unit triangular array with (n,k)-th entry (2^(n+1-k)-1)*binomial(n,k). For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0  M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. It follows from the recurrence equation given in the Formula section that the infinite product M(0)*M(1)*M(2)*..., which is clearly well-defined, is equal to the present triangle (but with the first row and column omitted). See the Example section. (End)
The Bell transform of 2^(n+1)-1. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			T(3,2) = 9: {1}{2}<-3, {1}{3}<-2, {1}{2,3}, {2}{1}<-3, {2}{3}<-1, {2}{1,3}, {3}{1}<-2, {3}{2}<-1, {3}{1,2}.
Triangle begins:
  1;
  0,   1;
  0,   3,    1;
  0,   7,    9,     1;
  0,  15,   55,    18,    1;
  0,  31,  285,   205,   30,    1;
  0,  63, 1351,  1890,  545,   45,  1;
  0, 127, 6069, 15421, 7770, 1190, 63,  1;
  ...
From _Peter Bala_, Jan 07 2015: (Start)
T(4,2) = 55: There are 7 partitions of the set {1,2,3,4} into 2 blocks. For the 3 set partitions of the type {a,b}{c,d} we can choose a nonempty subset from each block in one of 3*3 ways giving 3*3*3 = 27 possibilities in all. The remaining 4 set partitions of {1,2,3,4} into 2 blocks are of the form {a,b,c}{d} and we can choose a nonempty subset from each block in 7*1 ways giving 4*7*1 = 28 possible choices. Thus in total T(4,2) = 27 + 28 = 55.
Recurrence equation example:
T(4,2) = sum {j = 1..3} (2^(4-j) - 1)*binomial(3,j)*T(j,1) = 7*3*1 + 3*3*3 + 1*1*7 = 55.
Connection constants:
Row 3 = [0, 7, 9, 1]. Hence x^3 = 7*x + 9*x*(x - 3) + x*(x - 4)*(x - 5); Row 4 = [0, 15, 55, 18, 1]. Hence x^4 = 15*x + 55*x*(x - 3) + 18*x*(x - 4)*(x - 5) + x*(x - 5)*(x - 6)*(x - 7).
With the array M(k) as defined in the Comments section, the infinite product M(0)*M(1)*M(2)*... begins
/ 1        \/1           \/1       \       / 1        \
| 3  1     ||0  1        ||0 1      |      | 3  1      |
| 7  6  1  ||0  3  1     ||0 0 1    |... = | 7  9  1   |
|15 21 9 1 ||0  7  6  1  ||0 0 3 1  |      |15 55 18 1 |
|...       ||0 15 21  9 1||0 0 7 6 1|      |...        |
|...       ||...         ||...      |      |           |
(End)

Alois P. Heinz, Rows n = 0..140, flattened
Eli Bagno, Riccardo Biagioli, and David Garber, Some identities involving second kind Stirling numbers of types B and D, arXiv:1901.07830 [math.CO], 2019.
Peter Bala, A 3 parameter family of generalized Stirling numbers
Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.
Index entries for sequences related to rooted trees

Columns k=0-9: A000007, A000225, A016269(n-2), A028025(n-3), A143399, A143400, A143401, A143402, A143403, A143404.
Diagonal: A000012.
T(2*n,n) gives A383869.
See also A048993, A008277, A007318, A143405 for row sums.

[[(&+[Binomial(n,j)*StirlingSecond(j,k)*k^(n-j): j in [k..n]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Mar 07 2019

T:= (n, k)-> add(binomial(n,t)*Stirling2(t,k)*k^(n-t), t=k..n):
seq(seq(T(n, k), k=0..n), n=0..11);

t[0, 0]=1; t[n_, k_]:= SeriesCoefficient[Exp[y*Exp[x]*(Exp[x]-1)], {x, 0, n}, {y, 0, k}]*n!; Table[t[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Dec 05 2013, after Vladeta Jovovic *)
Table[If[n==k==0, 1, If[k==0, 0, Sum[Binomial[n, j]*StirlingS2[j, k]* k^(n-j), {j,k,n}]]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 07 2019 *)

{T(n,k) = sum(j=k, n, binomial(n,j)*stirling(j,k,2)*k^(n-j))};
for(n=0,10, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Mar 07 2019

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 2^(n+1)-1, 10) # Peter Luschny, Jan 18 2016

G.f. for column k: x^k/Product_{t=k..2*k} (1-t*x).
T(n,k) = Sum_{t=k..n} C(n,t) * Stirling2(t,k) * k^(n-t).
E.g.f.: exp(y*exp(x)*(exp(x)-1)). - Vladeta Jovovic, Dec 08 2008
T(n,k) = Sum_{m=0..k} Stirling2(n,k+m)*(k+m)!/(m!*(k-m)!). - Vladimir Kruchinin, Apr 06 2011
Let P be Pascal's triangle A007318. The first column of the array exp(t*(P^2-P)) gives the row generating polynomials of this triangle.
The row polynomials R(n,t) satisfy the recurrence R(n+1,t) = t*(Sum_{k = 0..n} (2^(k+1)-1)*C(n,k)*R(n-k,t)) with R(0,t) = 1. For example, the row 4 polynomial R(4,t) = 15*t + 55*t^2 + 18*t^3 + t^4 = t*((7*t + 9*t^2 + t^3) + 3*3*(3*t+t^2) + 7*3*t + 15*1). - Peter Bala, Oct 12 2011
From Peter Bala, Jan 07 2015: (Start)
T(n,k) = (1/k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*(j + k)^n.
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} (2^(n-j+1) - 1)*binomial(n,j)*T(j,k) with T(0,0) = 1 and T(n,0) = 0 for n >= 1. This leads to the matrix factorization noted in the Comments section.
The inverse array is a signed version of A038455. (End)

A000369 Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

Original entry on oeis.org

1, 3, 1, 21, 9, 1, 231, 111, 18, 1, 3465, 1785, 345, 30, 1, 65835, 35595, 7650, 825, 45, 1, 1514205, 848925, 196245, 24150, 1680, 63, 1, 40883535, 23586255, 5755050, 775845, 62790, 3066, 84, 1, 1267389585, 748471185, 190482705, 27478710

Offset: 1

Wolfdieter Lang

a(n,m) := S2p(-3; n,m), a member of a sequence of triangles including S2p(-1; n,m) := A001497(n-1,m-1) (Bessel triangle) and ((-1)^(n-m))*S2p(1; n,m) := A008277(n,m) (Stirling 2nd kind). a(n,1)= A008545(n-1).
a(n,m), n>=m>=1, enumerates unordered n-vertex m-forests composed of m increasing plane (aka ordered) trees, with one vertex of out-degree r=0 (leafs or a root) and each vertex with out-degree r>=1 comes in r+2 types (like for an (r+2)-ary vertex). Proof from the e.g.f. of the first column Y(z):=1-(1-4*x)^(1/4) and the F. Bergeron et al. reference given in A001498, eq. (8), Y'(z)= phi(Y(z)), Y(0)=0, with out-degree o.g.f. phi(w)=1/(1-w)^3. - Wolfdieter Lang, Oct 12 2007
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+3) (A008545) adding 1,0,0,0,... as column 0. For the definition of the Bell transform see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

			Triangle begins:
  1;
  3, 1;
  21, 9, 1;
  231, 111, 18, 1;
  3465, 1785, 345, 30, 1; ...
Tree combinatorics for a(3,2)=9: there are three m=2 forests each with one tree a root (with out-degree r=0) and the other tree a root and a leaf coming in three versions (like for a 3-ary vertex). Each such forest can be labeled increasingly in three ways (like (1,(23)), (2,(13)) and (3,(12))) yielding 9 such forests. - _Wolfdieter Lang_, Oct 12 2007

Vincenzo Librandi, Rows n = 1..50, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0212072, 2002.
Tom Copeland, A Class of Differential Operators and the Stirling Numbers
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, First ten rows.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Toufik Mansour, Matthias Schork and Mark Shattuck, The Generalized Stirling and Bell Numbers Revisited, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
Mathias Pétréolle, Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Index entries for sequences related to Bessel functions or polynomials

Row sums give A016036. Cf. A004747.
Columns include A008545.
Alternating row sums A132163.

a[n_, m_] /; n >= m >= 1 := a[n, m] = (4(n-1) - m)*a[n-1, m] + a[n-1, m-1]; a[n_, m_] /; n < m = 0; a[, 0] = 0; a[1, 1] = 1; Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]] (* _Jean-François Alcover, Jul 22 2011 *)

# uses[bell_transform from A264428]
# Adds a column 1,0,0,0,... at the left side of the triangle.
def A000369_row(n):
    multifact_4_3 = lambda n: prod(4*k + 3 for k in (0..n-1))
    mfact = [multifact_4_3(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A000369_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

a(n, m) = n!*A049213(n, m)/(m!*4^(n-m)); a(n+1, m) = (4*n-m)*a(n, m) + a(n, m-1), n >= m >= 1; a(n, m) := 0, n

E.g.f. of m-th column: ((1-(1-4*x)^(1/4))^m)/m!.

From Peter Bala, Jun 08 2016: (Start)

With offset 0, the e.g.f. is 1/(1 - 4*x)^(3/4)*exp(t*(1 - (1 - 4*x)^(1/4))) = 1 + (3 + t)*x + (21 + 9*t + t^2)*x^2/2! + ....

Thus with row and column numbering starting at 0, this triangle is the exponential Riordan array [d/dx(F(x)), F(x)], belonging to the Derivative subgroup of the exponential Riordan group, where F(x) = 1 - (1 - 4*x)^(1/4).

Row polynomial recurrence: R(n+1,t) = t*Sum_{k = 0..n} binomial(n,k)*A008545(k)*R(n-k,t) with R(0,t) = 1. (End)

A039810 Matrix square of Stirling2 triangle A008277: 2-levels set partitions of [n] into k first-level subsets.

Original entry on oeis.org

1, 2, 1, 5, 6, 1, 15, 32, 12, 1, 52, 175, 110, 20, 1, 203, 1012, 945, 280, 30, 1, 877, 6230, 8092, 3465, 595, 42, 1, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1, 115975, 2090424, 5971350, 5660615, 2350950, 487704, 53550, 3120, 90, 1

Offset: 1

Christian G. Bower, Feb 15 1999

This triangle groups certain generalized Stirling numbers of the second kind A000558, A000559, ... They can also be interpreted in terms of trees of height 3 with n leaves and constraints on the order of the root.
From Peter Bala, Jul 19 2014: (Start)
The (n,k)-th entry in this table gives the number of double partitions of the set [n] = {1,2,...,n} into k blocks. To form a double partition of [n] we first write [n] as a disjoint union X_1 U...U X_k of k nonempty subsets (blocks) X_i of [n]. Then each block X_i is further partitioned into sub-blocks to give a double partition. For instance, {1,2,4} U {3,5} is a partition of [5] into 2 blocks and {{1,4},{2}} U {{3},{5}} is a refinement of this partition to a double partition of [5] into 2 blocks (and 4 sub-blocks).
Compare the above interpretation for the (n,k)-th entry of this table with the interpretation of the (n,k)-th entry of A013609 (the square of Pascal's triangle but with the rows read in reverse order) as counting the pairs (X,Y) of subsets of [n] such that |Y| = k and X is contained in Y. (End)
Also the Bell transform of the shifted Bell numbers B(n+1) without column 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016
T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used and the colors are introduced in increasing order. T(3,2) = 6: 1a|23b, 13a|2b, 12a|3b, 1a|2a|3b, 1a|2b|3a, 1a|2b|3b. - Alois P. Heinz, Aug 27 2019

			Triangle begins:
      k = 1    2    3    4    5          sum
  n
  1       1                                1
  2       2    1                           3
  3       5    6    1                     12
  4      15   32   12    1                60
  5      52  175  110   20    1          358
Matrix multiplication Stirling2 * Stirling2:
                  1  0  0  0
                  1  1  0  0
                  1  3  1  0
                  1  7  6  1
.
  1  0  0  0      1  0  0  0
  1  1  0  0      2  1  0  0
  1  3  1  0      5  6  1  0
  1  7  6  1     15 32 12  1
From _Peter Bala_, Jul 19 2014: (Start)
T(5,2) = 175: A 5-set can be partitioned into 2 blocks as either a union of a 3-set and a 2-set or as a union of a 4-set and a singleton set.
In the first case there are 10 ways of partitioning a 5-set into a 3-set and a 2-set. Each 3-set can be further partitioned into sub-blocks in Bell(3) = 5 ways and each 2-set can be further partitioned into sub-blocks in Bell(2) = 2 ways. So altogether we obtain 10*5*2 = 100 double partitions of this type.
In the second case, there are 5 ways of partitioning a 5-set into a 4-set and a 1-set. Each 4-set can be further partitioned in Bell(4) = 15 ways and each 1-set can be further partitioned in Bell(1) = 1 way. So altogether we obtain 5*15*1 = 75 double partitions of this type.
Hence, in total, T(5,2) = 100 + 75 = 175. (End)

Tilman Piesk, First 100 rows, flattened
A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, Bell polynomials in combinatorial Hopf algebras, arXiv preprint arXiv:1402.2960 [math.CO], 2014-2015.
T. Mansour, A. Munagi, and M. Shattuck, Recurrence Relations and Two-Dimensional Set Partitions , J. Int. Seq. 14 (2011) # 11.4.1.

Cf. A039811, A039814, A039813 (other products of Stirling matrices).
T(n, 1) = A000110(n) (first column) (Bell numbers).
T(n, 2) = A000558(n) 2-levels set partitions with 2 first-level classes.
T(n, n-1) = A002378(n-1) = n*(n-1) = 2*C(n,2) = set-partitions into (n-2) singletons and one of the two possible set partitions of [2].
Sum is A000258(n), 2-levels set partitions.
Another version with offset 0: A130191.
Horizontal mirror triangle is A046817.
T(2n,n) gives A321712.

# The function BellMatrix is defined in A264428.
# Adds (1,0,0,0, ..) as column 0.
BellMatrix(n -> combinat:-bell(n+1), 10); # Peter Luschny, Jan 28 2016

Flatten[Table[Sum[StirlingS2[n,i]*StirlingS2[i,k],{i,k,n}],{n,1,10},{k,1,n}]] (* Indranil Ghosh, Feb 22 2017 *)
rows = 10;
t = Table[BellB[n+1], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

T(n, k) = sum(j=0, n, stirling(n, j, 2)*stirling(j, k, 2)); \\ Seiichi Manyama, Feb 13 2022

S2 = A008277 (Stirling numbers of the second kind).
T = (S2)^2.
T(n,k) = Sum_{i=k..n} S2(n,i) * S2(i,k).
E.g.f. of k-th column: (exp(exp(x)-1)-1)^k/k!. [corrected by Seiichi Manyama, Feb 12 2022]
From Peter Bala, Jul 19 2014: (Start)
T(n,k) = Sum_{disjoint unions X_1 U...U X_k = [n]} Bell(|X_1|)*...*Bell(|X_k|), where Bell(n) = A000110(n).
Recurrence equation: T(n+1,k+1) = Sum_{j = k..n} Bell(n+1-j)*binomial(n,j)* T(j,k).
Row sums [1,3,12,60,358,...] = A000258. (End)

Definition and interpretation edited by Olivier Gérard, Jul 31 2011

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A143543 Triangle read by rows: T(n,k) = number of labeled graphs on n nodes with k connected components, 1<=k<=n.

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A132056 Triangle read by rows, the Bell transform of Product_{k=0..n} 7*k+1 without column 0.

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A223511 Triangle T(n,k) represents the coefficients of (x^9*d/dx)^n, where n=1,2,3,...;generalization of Stirling numbers of second kind A008277, Lah-numbers A008297.

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A105599 Triangle read by rows: T(n, m) = number of forests with n nodes and m labeled trees. Also number of forests with exactly n - m edges on n labeled nodes.

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GAP

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PARI

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A122848 Exponential Riordan array (1, x(1+x/2)).

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A039683 Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).

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