A264428 -id:A264428 - cp's OEIS frontend

A111594 Triangle of arctanh numbers.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060524.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.
In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).
Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008
Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Binomial convolution of row polynomials:
p(3,x)= 2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1,
together with those from A060524:
s(3,x)= 5*x+x^3; s(2,x)= 1+x^2, s(1,x)= x, s(0,x)= 1; therefore:
5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = 2*y+y^3 + 3*x*y^2 + 3*(1+x^2)*y + (5*x+x^3).
Triangle begins:
  1;
  0,   1;
  0,   0,    1;
  0,   2,    0,   1;
  0,   0,    8,   0,    1;
  0,  24,    0,  20,    0,  1;
  0,   0,  184,   0,   40,  0,   1;
  0, 720,    0, 784,    0, 70,   0, 1;
  0,   0, 8448,   0, 2464,  0, 112, 0, 1;
...

Wolfdieter Lang, First 10 rows.

Row sums: A000246.

Cf. A005359, A049218, A060524, A111593.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016

rows = 10;
t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[riordan_array from A256893]
riordan_array(1, atanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((arctanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 26, 36, 12, 1, 0, 150, 250, 120, 20, 1, 0, 1082, 2040, 1230, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1

Offset: 0

Wolfdieter Lang, May 04 2007

Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011.
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,    6,    6,    1;
  0,   26,   36,   12,   1;
  0,  150,  250,  120,  20,  1;
  0, 1082, 2040, 1230, 300, 30,  1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.

Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.

Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.

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

rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)

def a_row(n):
    s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011

New name by Peter Luschny, Jun 27 2019

A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

Original entry on oeis.org

1, 0, 1, 0, -2, 1, 0, 9, -6, 1, 0, -64, 48, -12, 1, 0, 625, -500, 150, -20, 1, 0, -7776, 6480, -2160, 360, -30, 1, 0, 117649, -100842, 36015, -6860, 735, -42, 1, 0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1, 0, 43046721, -38263752, 14880348, -3306744, 459270, -40824, 2268, -72, 1

Offset: 0

Roger L. Bagula, Apr 18 2008

Row sums give A177885.
The Abel polynomials are associated with the Abel operator t*exp(y*t)*p(x) = t*p(x+y).
From Peter Luschny, Jan 14 2009: (Start)
Abs(T(n,k)) is the number of rooted labeled trees on n+1 vertices with a root degree k (Clarke's formula).
The row sums in the unsigned case, Sum_{k=0..n} abs(T(n,k)), count the trees on n+1 labeled nodes, A000272(n+1). (End)
Exponential Riordan array [1, W(x)], W(x) the Lambert W-function. - Paul Barry, Nov 19 2010
The inverse array is the exponential Riordan array [1, x*exp(x)], which is A059297. - Peter Bala, Apr 08 2013
The inverse Bell transform of [1,2,3,...]. See A264428 for the Bell transform and A264429 for the inverse Bell transform. - Peter Luschny, Dec 20 2015
Also the Bell transform of (-1)^n*(n+1)^n. - Peter Luschny, Jan 18 2016

			Triangle begins:
  1;
  0,        1;
  0,       -2,       1;
  0,        9,      -6,       1;
  0,      -64,      48,     -12,      1;
  0,      625,    -500,     150,    -20,      1;
  0,    -7776,    6480,   -2160,    360,    -30,    1;
  0,   117649, -100842,   36015,  -6860,    735,  -42,   1;
  0, -2097152, 1835008, -688128, 143360, -17920, 1344, -56, 1;

Steve Roman, The Umbral Calculus, Dover Publications, New York (1984), pp. 14 and 29

Seiichi Manyama, Rows n = 0..139, flattened
W. Y. Chen, A general bijective algorithm for trees, PNAS December 1, 1990 vol. 87 no. 24 9635-9639.
L. E. Clarke, On Cayley's formula for counting trees, J. London Math. Soc. 33 (1958), 471-475.
Péter L. Erdős and L. A. Székely, Applications of Antilexicographic Order. I., An Enumerative Theory of Trees, Adv. in Appl. Math. 10, (1989) 488-496.
Eric Weisstein's World of Mathematics, Abel Polynomial.
Wikipedia, Abel Polynomials.
Bao-Xuan Zhu, Total positivity from a generalized cycle index polynomial, arXiv:2006.14485 [math.CO], 2020.

Row sums A177885.
Cf. A000272, A061356, A059297 (inverse array), A264429.
Cf. A061354, A095996, A264234, A061355, A049444.

T := proc(n,k) if n = 0 and k = 0 then 1 else binomial(n-1,k-1)*(-n)^(n-k) fi end; seq(print(seq(T(n,k),k=0..n)),n=0..7); # Peter Luschny, Jan 14 2009
# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-n-1)^n, 9); # Peter Luschny, Jan 27 2016

a0 = 1 a[x, 0] = 1; a[x, 1] = x; a[x_, n_] := x*(x - a0*n)^(n - 1); Table[Expand[a[x, n]], {n, 0, 10}]; a1 = Table[CoefficientList[a[x, n], x], {n, 0, 10}]; Flatten[a1]
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, (-n-1)^n], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

# uses[inverse_bell_transform from A264429]
def A137452_matrix(dim):
    nat = [n for n in (1..dim)]
    return inverse_bell_transform(dim, nat)
A137452_matrix(10) # Peter Luschny, Dec 20 2015

Row n gives the coefficients of the expansion of x*(x-n)^(n-1).
Abs(T(n,k)) = C(n-1,k-1)*n^(n-k). - Peter Luschny, Jan 14 2009
From Wolfdieter Lang, Nov 08 2022: (Start)
From the exponential Riordan (also Sheffer of Jabotinsky) type (1, LambertW) array (see comments).
E.g.f. of column sequence k, LambertW(x)^k/k!, for k >= 0.
E.g.f. of row polynomials P_n(y) = Sum_{k=0..n} T(n, k)*y^k: exp(y*LambertW(x)).
Recurrence for T: T(n, k) = 0 for n < k; T(n, 0) = 1 for n = 0 otherwise 0; T(n, k) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j,k-1)*(-1)^j*T(n-1, k-1+j). (Jabotinsky type convolution triangle, the e.g.f.s for the a- and z-sequences are exp(-x), and 0. See the link in A006232.)
Recurrence for column k of T: T(n, k) = 0 for n < k, T(k, k) = 1, for k >= 0 otherwise T(n, k) = (n!*k/(n-k))*Sum_{j=k..n-1} (1/j!)*beta(n-1-j)*T(j, k), where beta(n) = A264234(n+1)/A095996(n+1) = {-1, 2, -9/2, 32/3, -625/24, ...} with o.g.f. d/dx(log(LambertW(x)/x)). See the Boas-Buck or Rainville references given in A046521, and my Aug 10 2017 comment there.
Recurrence for the row polynomials P_0(x) = 1, and P_n(x) = x*substitute(z=d/dx, exp(-z)/(1+z)) P_(n-1)(x), for n >= 1, with coefficient z^k of exp(-z)/(1+z) given by (-1)^k*A061354(k)/A061355(k). See the Roman reference Corollary 3.7.2., p. 50. (End)
The column sequences for the unsigned triangle Abs(T(n, k)), for k >= 2, are also given by {n^(n-k)*(n-1)*s(k-2, n)/(k-1)!}A049444.%20-%20_Wolfdieter%20Lang">{n>=k} with the row polynomials s(n, x) = risingfactorial(x - (n+1), n) of A049444. - _Wolfdieter Lang, Nov 21 2022

Better name by Peter Bala, Apr 08 2013

Edited by Joerg Arndt, Apr 08 2013

A147309 Riordan array [sec(x), log(sec(x) + tan(x))].

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 4, 0, 1, 5, 0, 10, 0, 1, 0, 40, 0, 20, 0, 1, 61, 0, 175, 0, 35, 0, 1, 0, 768, 0, 560, 0, 56, 0, 1, 1385, 0, 4996, 0, 1470, 0, 84, 0, 1, 0, 24320, 0, 22720, 0, 3360, 0, 120, 0, 1

Offset: 0

Paul Barry, Nov 05 2008

Production array is [cosh(x),x] beheaded. Inverse is A147308. Row sums are A000111(n+1).
Unsigned version of A147308. - N. J. A. Sloane, Nov 07 2008
From Peter Bala, Jan 26 2011: (Start)
Define a polynomial sequence {Z(n,x)} n >= 0 by means of the recursion
(1)... Z(n+1,x) = 1/2*x*{Z(n,x-1)+Z(n,x+1)}
with starting condition Z(0,x) = 1. We call Z(n,x) the zigzag polynomial of degree n. This table lists the coefficients of these polynomials (for n >= 1) in ascending powers of x, row indices shifted by 1. The first few polynomials are
... Z(1,x) = x
... Z(2,x) = x^2
... Z(3,x) = x + x^3
... Z(4,x) = 4*x^2 + x^4
... Z(5,x) = 5*x + 10*x^3 + x^5.
The value Z(n,1) equals the zigzag number A000111(n). The polynomials Z(n,x) occur in formulas for the enumeration of permutations by alternating descents A145876 and in the enumeration of forests of non-plane unary binary labeled trees A147315.
{Z(n,x)}n>=0 is a polynomial sequence of binomial type and so is analogous to the sequence of monomials x^n. Denoting Z(n,x) by x^[n] to emphasize this analogy, we have, for example, the following analog of Bernoulli's formula for the sum of integer powers:
(2)... 1^[m]+...+(n-1)^[m] = (1/(m+1))*Sum_{k=0..m} (-1)^floor(k/2)*binomial(m+1,k)*B_k*n^[m+1-k],
where {B_k} k >= 0 = [1, -1/2, 1/6, 0, -1/30, ...] is the sequence of Bernoulli numbers.
For similarly defined polynomial sequences to Z(n,x) see A185415, A185417 and A185419. See also A185424.
(End)
[gd(x)^(-1)]^m = Sum_{n>=m} Tg(n,m)*(m!/n!)*x^n, where gd(x) is Gudermannian function, Tg(n+1,m+1)=T(n,m). - Vladimir Kruchinin, Dec 18 2011
The Bell transform of abs(E(n)), E(n) the Euler numbers. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
   1;
   0,  1;
   1,  0,   1;
   0,  4,   0,  1;
   5,  0,  10,  0,  1;
   0, 40,   0, 20,  0, 1;
  61,  0, 175,  0, 35, 0, 1;

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Cf. A000111 (row sums), A000364, A012123, A012259, A145876, A147315, A185415, A185417, A185419, A185421, A185424. - Peter Bala, Jan 26 2011

Z := proc(n, x) option remember;
description 'zigzag polynomials Z(n, x)'
if n = 0 return 1 else return 1/2*x*(Z(n-1, x-1)+Z(n-1, x+1)) end proc:
with(PolynomialTools):
for n from 1 to 10 CoefficientList(Z(n, x), x); end do; # Peter Bala, Jan 26 2011

t[n_, k_] := SeriesCoefficient[ 2^k*ArcTan[(E^x - 1)/(E^x + 1)]^k*n!/k!, {x, 0, n}]; Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten // Abs (* Jean-François Alcover, Jan 23 2015 *)

T(n, k)=local(X); if(k<1 || k>n, 0, X=x+x*O(x^n); n!*polcoeff(polcoeff((tan(X)+1/cos(X))^y, n), k)) \\ Paul D. Hanna, Feb 06 2011

R = PolynomialRing(QQ, 'x')
@CachedFunction
def zzp(n, x) :
    return 1 if n == 0 else x*(zzp(n-1, x-1)+zzp(n-1, x+1))/2
def A147309_row(n) :
    x = R.gen()
    L = list(R(zzp(n, x)))
    del L[0]
    return L
for n in (1..10) : print(A147309_row(n)) # Peter Luschny, Jul 22 2012

# uses[bell_matrix from A264428]
# Alternative: Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: abs(euler_number(n)), 10) # Peter Luschny, Jan 18 2016

From Peter Bala, Jan 26 2011: (Start)
GENERATING FUNCTION
The e.g.f., upon including a constant term of '1', is given by:
(1) F(x,t) = (tan(t) + sec(t))^x = Sum_{n>=0} Z(n,x)*t^n/n! = 1 + x*t + x^2*t^2/2! + (x+x^3)*t^3/3! + ....
Other forms include
(2) F(x,t) = exp(x*arcsinh(tan(t))) = exp(2*x*arctanh(tan(t/2))).
(3) F(x,t) = exp(x*(t + t^3/3! + 5*t^5/5! + 61*t^7/7! + ...)),
where the coefficients [1,1,5,61,...] are the secant or zig numbers A000364.
ROW GENERATING POLYNOMIALS
One easily checks from (1) that
d/dt(F(x,t)) = 1/2*x*(F(x-1,t) + F(x+1,t))
and so the row generating polynomials Z(n,x) satisfy the recurrence relation
(4) Z(n+1,x) = 1/2*x*{Z(n,x-1) + Z(n,x+1)}.
The e.g.f. for the odd-indexed row polynomials is
(5) sinh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n+1,x)*t^(2n+1)/(2n+1)!.
The e.g.f. for the even-indexed row polynomials is
(6) cosh(x*arcsinh(tan(t))) = Sum_{n>=0} Z(2n,x)*t^(2n)/(2n)!.
From sinh(2*x) = 2*sinh(x)*cosh(x) we obtain the identity
(7) Z(2n+1,2*x) = 2*Sum_{k=0..n} binomial(2n+1,2k)*Z(2k,x)*Z(2n-2k+1,x).
The zeros of Z(n,x) lie on the imaginary axis (use (4) and adapt the proof given in A185417 for the zeros of the polynomial S(n,x)).
BINOMIAL EXPANSION
The form of the e.g.f. shows that {Z(n,x)} n >= 0 is a sequence of polynomials of binomial type. In particular, we have the expansion
(8) Z(n,x+y) = Sum_{k=0..n} binomial(n,k)*Z(k,x)*Z(n-k,y).
The delta operator D* associated with this binomial type sequence is
(9) D* = D - D^3/3! + 5*D^5/5! - 61*D^7/7! + 1385*D^9/9! - ..., and satisfies
the relation
(10) tan(D*)+sec(D*) = exp(D).
The delta operator D* acts as a lowering operator on the zigzag polynomials:
(11) (D*)Z(n,x) = n*Z(n-1,x).
ANALOG OF THE LITTLE FERMAT THEOREM
For integer x and odd prime p
(12) Z(p,x) = (-1)^((p-1)/2)*x (mod p).
More generally, for k = 1,2,3,...
(13) Z(p+k-1,x) = (-1)^((p-1)/2)*Z(k,x) (mod p).
RELATIONS WITH OTHER SEQUENCES
Row sums [1,1,2,5,16,61,...] are the zigzag numbers A000111(n) for n >= 1.
Column 1 (with 0's omitted) is the sequence of Euler numbers A000364.
A145876(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n+1,k-j)*Z(n,j).
A147315(n-1,k-1) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).
A185421(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j).
A012123(n) = (-i)^n*Z(n,i) where i = sqrt(-1). A012259(n) = 2^n*Z(n,1/2).
(End)
T(n,m) = Sum(i=0..n-m, s(i)/(n-i)!*Sum(k=m..n-i, A147315(n-i,k)*Stirling1(k,m))), m>0, T(n,0) = s(n), s(n)=[1,0,1,0,5,0,61,0,1385,0,50521,...] (see A000364). - Vladimir Kruchinin, Mar 10 2011

A147315 L-matrix for Euler numbers A000111(n+1).

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 11, 6, 1, 16, 45, 35, 10, 1, 61, 211, 210, 85, 15, 1, 272, 1113, 1351, 700, 175, 21, 1, 1385, 6551, 9366, 5901, 1890, 322, 28, 1, 7936, 42585, 70055, 51870, 20181, 4410, 546, 36, 1, 50521, 303271, 563970, 479345, 218925, 58107, 9240, 870, 45, 1

Offset: 0

Paul Barry, Nov 05 2008

This is the inverse of the coefficient array for the orthogonal polynomials p(n,x) defined by: p(n,x)=if(n=-1,0,if(n=0,1,(x-n)p(n-1,x)-C(n,2)p(n-2,x))).
The Hankel array H for A000111(n+1) satisfies H=L*D*U with U the transpose of L.
Row sums are A000772(n+1) with e.g.f. dif(exp(-1)exp(sec(x)+tan(x)),x).
From Peter Bala, Jan 31 2011: (Start)
The following comments refer to the table with an offset of 1: i.e., both the row and column indexing starts at 1.
An increasing tree is a labeled rooted tree with the property that the sequence of labels along any path starting from the root is increasing. A000111(n) for n>=1 enumerates the number of increasing unordered trees on the vertex set {1,2,...,n}, rooted at 1, in which all outdegrees are <=2 (plane unary-binary trees in the notation of [Bergeron et al.])
The entry T(n,k) of the present table gives the number of forests of k increasing unordered trees on the vertex set {1,2,...,n} in which all outdegrees are <=2. See below for some examples.
For ordered forests of such trees see A185421. For forests of increasing ordered trees on the vertex set {1,2,...,n}, rooted at 1, in which all outdegrees are <=2, see A185422.
The Stirling number of the second kind Stirling2(n,k) is the number of partitions of the set [n] into k blocks. Arranging the elements in each block in ascending numerical order provides an alternative combinatorial interpretation for Stirling2(n,k) as counting forests of k increasing unary trees on n nodes. Thus we may view the present array, which counts increasing unary-binary trees, as generalized Stirling numbers of the second kind associated with A000111 or with the zigzag polynomials Z(n,x) of A147309 - see especially formulas (2) and (3) below.
See A145876 for generalized Eulerian numbers associated with A000111. (End)
The Bell transform of A000111(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 18 2016

			Triangle begins
    1;
    1,    1;
    2,    3,    1;
    5,   11,    6,   1;
   16,   45,   35,  10,   1;
   61,  211,  210,  85,  15,  1;
  272, 1113, 1351, 700, 175, 21, 1;
  ...
The production array for L is the tridiagonal array
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  0,  0,  6,  4,  1;
  0,  0,  0, 10,  5,  1;
  0,  0,  0,  0, 15,  6,  1;
  0,  0,  0,  0,  0, 21,  7,  1;
  0,  0,  0,  0,  0,  0, 28,  8,  1,;
  0,  0,  0,  0,  0,  0,  0, 36,  9,  1;
From _Peter Bala_, Jan 31 2011: (Start)
Examples of forests:
The diagrams below are drawn so that the leftmost child of a binary node has the maximum label.
T(4,1) = 5. The 5 forests consisting of a single non-plane increasing unary-binary tree on 4 nodes are
...4... ........ .......... ........... ...........
...|... ........ .......... ........... ...........
...3... .4...3.. .4........ ........4.. ........3..
...|... ..\./... ..\....... ......./... ......./...
...2... ...2.... ...3...2.. ..3...2.... ..4...2....
...|... ...|.... ....\./... ...\./..... ...\./.....
...1... ...1.... .....1.... ....1...... ....1......
T(4,2) = 11. The 11 forests consisting of two non-plane increasing unary-binary trees on 4 nodes are
......... ...3.....
.3...2... ...|.....
..\./.... ...2.....
...1...4. ...|.....
......... ...1...4.
.
......... ...4.....
.4...2... ...|.....
..\./.... ...2.....
...1...3. ...|.....
......... ...1...3.
.
......... ...4.....
.4...3... ...|.....
..\./.... ...3.....
...1...2. ...|.....
......... ...1...2.
.
......... ...4.....
.4...3... ...|.....
..\./.... ...3.....
...2...1. ...|.....
......... ...2...1.
.
......... ......... ..........
..2..4... ..3..4... ..4...3...
..|..|... ..|..|... ..|...|...
..1..3... ..1..2... ..1...2...
......... ......... .......... (End)

Alois P. Heinz, Rows n = 0..140, flattened
Paul Barry, A Note on Three Families of Orthogonal Polynomials defined by Circular Functions, and Their Moment Sequences, Journal of Integer Sequences, Vol. 15 (2012), #12.7.2.
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.
Tom Copeland, Mathemagical Forests
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Cf. A145876, A147309, A185421, A185422, A185424.

A147315 := proc(n,k) n!*exp(x*(sec(t)+tan(t)-1)) - 1: coeftayl(%,t=0,n) ; coeftayl(%,x=0,k) ; end proc:
seq(seq(A147315(n,k),k=1..n),n=0..12) ; # R. J. Mathar, Mar 04 2011
# second Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
g:= proc(n) option remember; expand(`if`(n=0, 1, add(
      g(n-j)*x*binomial(n-1, j-1)*b(j, 0), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n+1))(g(n+1)):
seq(T(n), n=0..10);  # Alois P. Heinz, May 19 2021

t[n_, k_] := t[n, k] = t[n-1, k-1] + (k+1)*t[n-1, k] + 1/2*(k+1)*(k+2)*t[n-1, k+1]; t[n_, k_] /; (n < 0 || k < 0 || k > n) = 0; t[0, 0] = t[1, 0] = 1; Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-François Alcover, Jun 21 2011, after PARI prog. *)

Co(n,k):=sum(binomial(k,j)*(if oddp(n-k+j) then 0 else if (n-k+j)/2A147315(n,m):=1/m!*sum((if oddp(n-k) then 0 else 2^(1-k)*sum((-1)^(floor((n+k)/2)-i)*binomial(k,i)*(2*i-k)^n,i,0,floor(k/2)))*(sum(Co(i,m)*binomial(k-i+m-1,m-1),i,1,k)),k,m,n); /* Vladimir Kruchinin, Feb 17 2011 */

T(n,m):=(sum(binomial(k+m,m)*((-1)^(n-k-m)+1)*sum(binomial(j+k+m,k+m)*(j+k+m+1)!*2^(-j-k-1)*(-1)^((n+k+m)/2+j+k+m)*stirling2(n+1,j+k+m+1), j,0,n-k-m), k,0,n-m))/(m+1)!; /* Vladimir Kruchinin, May 17 2011 */

{T(n,k)=if(k<0||k>n,0,if(n==0,1,T(n-1,k-1)+(k+1)*T(n-1,k)+(k+1)*(k+2)/2*T(n-1,k+1)))} /* offset=0 */

{T(n,k)=local(X=x+x*O(x^(n+2)));(n+1)!*polcoeff(polcoeff(exp(y*((1+sin(X))/cos(X)-1))-1,n+1,x),k+1,y)} /* offset=0 */

/* Generate from the production matrix P: */
{T(n,k)=local(P=matrix(n,n,r,c,if(r==c-1,1,if(r==c,c,if(r==c+1,c*(c+1)/2)))));if(k<0||k>n,0,if(n==k,1,(P^n)[1,k+1]))}

# uses[bell_matrix from A264428, A000111]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: A000111(n+1), 10) # Peter Luschny, Jan 18 2016

From Peter Bala, Jan 31 2011: (Start)
The following formulas refer to the table with an offset of 1: i.e., both the row n and column k indexing start at 1.
GENERATING FUNCTION
E.g.f.:
(1)... exp(x*(sec(t)+tan(t)-1)) - 1 = Sum_{n>=1} R(n,x)*t^n/n!
= x*t + (x+x^2)*t^2/2! + (2*x+3*x^2+x^3)*t^3/3! + ....
TABLE ENTRIES
(2)... T(n,k) = (1/k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*Z(n,j),
where Z(n,x) denotes the zigzag polynomials as described in A147309.
Compare (2) with the formula for the Stirling numbers of the second kind
(3)... Stirling2(n,k) = (1/k!)*Sum_{j = 0..k} (-1)^(k-j)*binomial(k,j)*j^n.
Recurrence relation
(4)... T(n+1,k) = T(n,k-1) + k*T(n,k) + (1/2)*k(k+1)*T(n,k+1).
ROW POLYNOMIALS
The row polynomials R(n,x) begin
R(1,x) = x
R(2,x) = x+x^2
R(3,x) = 2*x+3*x^2+x^3
They satisfy the recurrence
(5)... R(n+1,x) = x*{R(n,x)+R'(n,x) + (1/2)*R''(n,x)},
where ' indicates differentiation with respect to x. This should be compared with the recurrence satisfied by the Bell polynomials Bell(n,x)
(6)... Bell(n+1,x) = x*(Bell(n,x) + Bell'(n,x)). (End)
From Vladimir Kruchinin, Feb 17 2011: (Start)
Sum_{m=1..n} T(n,m) = A000772(n).
Sum_{m=1..2n-1} T(2n-1,m)* Stirling1(m,1) = A000364(n).
Let Co(n,k) = Sum_{j=1..k} binomial(k,j)*(if (n-k+j) is odd then 0 else if (n-k+j)/2T(n,m) = m!* Sum_{k=m..n} (if n-k is odd then 0 else 2^(1-k)) * Sum_{i=0..floor(k/2)} (-1)^(floor((n+k)/2)-i) * binomial(k,i) * (2*i-k)^n)))) * Sum_{i=1..k} Co(i,m) * binomial(k-i+m-1,m-1), n>0.
(End)
T(n,m) = Sum_{k = 0..n-m} binomial(k+m,m)*((-1)^(n-k-m)+1)*Sum_{j=0..n-k-m} binomial(j+k+m,k+m)*(j+k+m+1)!*2^(-j-k-1)*(-1)^((n+k+m)/2+j+k+m)* Stirling2(n+1,j+k+m+1)/(m+1)!. - Vladimir Kruchinin, May 17 2011
The row polynomials R(n,x) are given by D^n(exp(x*t)) evaluated at t = 0, where D is the operator (1+t+t^2/2!)*d/dt. Cf. A008277 and A094198. See also A185422. - Peter Bala, Nov 25 2011

More terms from Michel Marcus, Mar 01 2014

A130191 Square of the Stirling2 matrix A048993.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 5, 6, 1, 0, 15, 32, 12, 1, 0, 52, 175, 110, 20, 1, 0, 203, 1012, 945, 280, 30, 1, 0, 877, 6230, 8092, 3465, 595, 42, 1, 0, 4140, 40819, 70756, 40992, 10010, 1120, 56, 1, 0, 21147, 283944, 638423, 479976, 156072, 24570, 1932, 72, 1

Offset: 0

Wolfdieter Lang, Jun 01 2007

Without row n=0 and column k=0 this is triangle A039810.
This is an associated Sheffer matrix with e.g.f. of the m-th column ((exp(f(x))-1)^m)/m! with f(x)=:exp(x)-1.
The triangle is also called the exponential Riordan array [1, exp(exp(x)-1)]. - Peter Luschny, Apr 19 2015
Also the Bell transform of shifted Bell numbers A000110(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle starts:
  1;
  0,   1;
  0,   2,    1;
  0,   5,    6,    1;
  0,  15,   32,   12,    1;
  0,  52,  175,  110,   20,   1;
  0, 203, 1012,  945,  280,  30,  1;
  0, 877, 6230, 8092, 3465, 595, 42, 1;

G. C. Greubel, Rows n=0..100 of triangle, flattened
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First 10 rows and more
John Riordan, Letter, Apr 28 1976. (See third page)

Columns k=0..3 give A000007, A000110 (for n > 0), A000558, A000559.
Row sums: A000258.
Alternating row sums: A130410.
T(2n,n) gives A321712.
Cf. A039810 (another version), A048993.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> combinat:-bell(n+1), 9); # Peter Luschny, Jan 27 2016

BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
rows = 10;
M = BellMatrix[BellB[# + 1]&, rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
a[n_, m_]:= Sum[StirlingS2[n, k]*StirlingS2[k, m], {k,m,n}]; Table[a[n, m], {n, 0, 100}, {m, 0, n}]//Flatten (* G. C. Greubel, Jul 10 2018 *)

for(n=0, 9, for(k=0, n, print1(sum(j=k, n, stirling(n, j, 2)*stirling(j, k, 2)), ", "))) \\ G. C. Greubel, Jul 10 2018

# uses[riordan_array from A256893]
riordan_array(1, exp(exp(x) - 1), 8, exp=true) # Peter Luschny, Apr 19 2015

a(n,k) = Sum_{j=k..n} S2(n,j) * S2(j,k), n>=k>=0.
E.g.f. row polynomials with argument x: exp(x*f(f(z))).
E.g.f. column k: ((exp(exp(x) - 1) - 1)^k)/k!.

A008298 Triangle of D'Arcais numbers.

Original entry on oeis.org

1, 3, 1, 8, 9, 1, 42, 59, 18, 1, 144, 450, 215, 30, 1, 1440, 3394, 2475, 565, 45, 1, 5760, 30912, 28294, 9345, 1225, 63, 1, 75600, 293292, 340116, 147889, 27720, 2338, 84, 1, 524160, 3032208, 4335596, 2341332, 579369, 69552, 4074, 108, 1, 6531840, 36290736, 57773700, 38049920, 11744775, 1857513, 154350, 6630, 135, 1

Offset: 1

N. J. A. Sloane

Also the Bell transform of A038048(n+1) and the inverse Bell transform of A180563(n+1) (adding 1,0,0,.. as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

Named after the Italian mathematician Francesco Flores D'Arcais (1849-1927). - Amiram Eldar, Jun 13 2021

			exp(Sum_{n>0} sigma(n)*u*x^n/n) = 1+u*x/1!+(3*u+u^2)*x^2/2!+(8*u+9*u^2+u^3)*x^3/3!+(42*u+59*u^2+18*u^3+u^4)*x^4/4!+...
Triangle starts:
      1:
      3,      1;
      8,      9,      1;
     42,     59,     18,      1;
    144,    450,    215,     30,     1;
   1440,   3394,   2475,    565,    45,    1;
   5760,  30912,  28294,   9345,  1225,   63,  1;
  75600, 293292, 340116, 147889, 27720, 2338, 84, 1;
  ...
T(4; u) = 4!*(binomial(u+3,4) + binomial(u+1,2)*binomial(u,1) + binomial(u+1,2) + binomial(u,1)^2 + binomial(u,1)) = 42*u+59*u^2+18*u^3+u^4.

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.
F. D'Arcais, Développement en série, Intermédiaire Math., Vol. 20 (1913), pp. 233-234.

Seiichi Manyama, Rows n = 1..100, flattened (rows n = 1..20 from Vincenzo Librandi)
Peter Luschny, The Bell transform.

Column k=1..3 give A038048, A059356, A059357.
Row sums give A053529.
Cf. A075525, A180563, A210590.

P := proc(n): if n=0 then 1 else P(n):= (1/n)*(add(x(n-k) * P(k), k=0..n-1)) fi; end: with(numtheory): x := proc(n): sigma(n) * x end: Q := proc(n): n!*P(n) end: T := proc(n, k): coeff(Q(n), x, k) end: seq(seq(T(n, k), k=1..n), n=1..10); # Johannes W. Meijer, Jul 08 2016

t[0][u_] = 1; t[n_][u_] := t[n][u] = Sum[(n-1)!/(n-k)!*DivisorSigma[1, k]*u*t[n-k][u], {k, 1, n}]; row[n_] := CoefficientList[ t[n][u], u] // Rest; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Oct 03 2012, after Vladeta Jovovic *)

row(n)={local(P(n)=if(n,sum(k=0,n-1,sigma(n-k)*x*P(k))/n,1)); Vecrev(P(n)*n!/x)} \\ T(n,k)=row(n)[k]. - M. F. Hasler, Jul 13 2016

a(n) = if(n<1, 0, (n-1)!*sigma(n));
T(n, k) = if(k==0, 0^n, sum(j=0, n-k+1, binomial(n-1, j-1)*a(j)*T(n-j, k-1))) \\ Seiichi Manyama, Nov 08 2020 after Peter Luschny

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
print(bell_matrix(lambda n: A038048(n+1), 9)) # Peter Luschny, Jan 19 2016

G.f.: Sum_{1<=k<=n} T(n, k)*u^k*t^n/n! = ((1-t)*(1-t^2)*(1-t^3)...)^(-u).
Recurrence for degree n D'Arcais polynomials T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = Sum_{k=1..n} (n-1)!/(n-k)!*sigma(k)*u*T(n-k; u), T(0; u) = 1. - Vladeta Jovovic, Oct 11 2002
T(n; u) = n!*Sum_{pi} Product_{i=1..n} binomial(u+k(i)-1, k(i)) where pi runs through all nonnegative solutions of k(1)+2*k(2)+..+n*k(n)=n. - Vladeta Jovovic, Oct 11 2002
E.g.f.: exp(Sum_{n>0} sigma(n)*u*x^n/n), where sigma(n)=A000203(n). - Vladeta Jovovic, Jan 10 2003
T(n, k) = coeff(n!*P(n), x^k), n >= 1 and 1 <= k <= n, with P(n) = (1/n)*Sum_{k=0..n-1} sigma(n-k)*P(k)*x for n >= 1 and P(n=0) = 1. See A036039. - Johannes W. Meijer, Jul 08 2016
T(n, k) = (n!/k!) * Sum_{i_1,i_2,...,i_k > 0 and i_1+i_2+...+i_k=n} Product_{j=1..k} sigma(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

More terms from Vladeta Jovovic, Dec 28 2001

A013988 Triangle read by rows, the inverse Bell transform of n!*binomial(5,n) (without column 0).

Original entry on oeis.org

1, 5, 1, 55, 15, 1, 935, 295, 30, 1, 21505, 7425, 925, 50, 1, 623645, 229405, 32400, 2225, 75, 1, 21827575, 8423415, 1298605, 103600, 4550, 105, 1, 894930575, 358764175, 59069010, 5235405, 271950, 8330, 140, 1, 42061737025, 17398082625, 3016869625, 289426830, 16929255, 621810, 14070, 180, 1

Offset: 1

Wolfdieter Lang

Previous name was: Triangle of numbers related to triangle A049224; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.
T(n, m) = S2p(-5; 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). T(n, 1) = A008543(n-1).
For the definition of the Bell transform see A264428 and the link. - Peter Luschny, Jan 16 2016

			Triangle begins as:
          1;
          5,         1;
         55,        15,        1;
        935,       295,       30,       1;
      21505,      7425,      925,      50,      1;
     623645,    229405,    32400,    2225,     75,     1;
   21827575,   8423415,  1298605,  103600,   4550,   105,    1;
  894930575, 358764175, 59069010, 5235405, 271950,  8330,  140,   1;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
Peter Luschny, The Bell transform
Index entries for sequences related to Bessel functions or polynomials

Cf. A008277, A008543, A049224, A264428.
Cf. A028844 (row sums).
Triangles with the recurrence T(n,k) = (m*(n-1)-k)*T(n-1,k) + T(n-1,k-1): A010054 (m=1), A001497 (m=2), A004747 (m=3), A000369 (m=4), A011801 (m=5), this sequence (m=6).

function T(n,k) // T = A013988
  if k eq 0 then return 0;
  elif k eq n then return 1;
  else return (6*(n-1)-k)*T(n-1,k) + T(n-1,k-1);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2023

(* First program *)
rows = 10;
b[n_, m_] := BellY[n, m, Table[k! Binomial[5, k], {k, 0, rows}]];
A = Table[b[n, m], {n, 1, rows}, {m, 1, rows}] // Inverse // Abs;
A013988 = Table[A[[n, m]], {n, 1, rows}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k==0, 0, If[k==n, 1, (6*(n-1) -k)*T[n-1,k] +T[n-1, k-1]]];
Table[T[n,k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Oct 03 2023 *)

# uses[inverse_bell_matrix from A264428]
# Adds 1,0,0,0, ... as column 0 at the left side of the triangle.
inverse_bell_matrix(lambda n: factorial(n)*binomial(5, n), 8) # Peter Luschny, Jan 16 2016

T(n, m) = n!*A049224(n, m)/(m!*6^(n-m));

T(n+1, m) = (6*n-m)*T(n, m) + T(n, m-1), for n >= m >= 1, with T(n, m) = 0, n

E.g.f. of m-th column: ((1 - (1-6*x)^(1/6))^m)/m!.

Sum_{k=1..n} T(n, k) = A028844(n).

New name from Peter Luschny, Jan 16 2016

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A075498 Stirling2 triangle with scaled diagonals (powers of 3).

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A075499 Stirling2 triangle with scaled diagonals (powers of 4).

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A111594 Triangle of arctanh numbers.

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A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A137452 Triangular array of the coefficients of the sequence of Abel polynomials A(n,x) := x*(x-n)^(n-1).

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A147309 Riordan array [sec(x), log(sec(x) + tan(x))].

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