A122848 -id:A122848 - cp's OEIS frontend

A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

1, -1, 1, 0, -3, 1, 0, 3, -6, 1, 0, 0, 15, -10, 1, 0, 0, -15, 45, -15, 1, 0, 0, 0, -105, 105, -21, 1, 0, 0, 0, 105, -420, 210, -28, 1, 0, 0, 0, 0, 945, -1260, 378, -36, 1, 0, 0, 0, 0, -945, 4725, -3150, 630, -45, 1, 0, 0, 0, 0, 0, -10395, 17325, -6930, 990, -55, 1, 0, 0, 0, 0, 0, 10395, -62370, 51975, -13860, 1485, -66, 1

Offset: 0

Paul D. Hanna, Mar 14 2005

Exponential Riordan array [1 - x, x - x^2/2]; cf. A049403. - Peter Bala, Apr 08 2013

Also the Bell transform of (-1)^n if n<2 else 0 and the inverse Bell transform of A001147(n) (adding 1,0,0,... as column 0). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Rows begin:
   1;
  -1,  1;
   0, -3,   1;
   0,  3,  -6,    1;
   0,  0,  15,  -10,    1;
   0,  0, -15,   45,  -15,     1;
   0,  0,   0, -105,  105,   -21,     1;
   0,  0,   0,  105, -420,   210,   -28,   1;
   0,  0,   0,    0,  945, -1260,   378, -36,   1;
   0,  0,   0,    0, -945,  4725, -3150, 630, -45, 1; ...
The columns being equal in absolute value to the rows of the matrix inverse A001497:
    1;
    1,   1;
    3,   3,   1;
   15,  15,   6,   1;
  105, 105,  45,  10,  1;
  945, 945, 420, 105, 15, 1; ...

G. C. Greubel, Rows n=0..35 of triangle, flattened
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]

Row sums are found in A001464 (offset 1).
Absolute row sums equal A000085.
Cf. A001497, A049403, A096713, A122848, A130757.

With[{nmax = 10}, CoefficientList[CoefficientList[Series[(1 - t)*Exp[x*(t - t^2/2)], {t, 0, nmax}, {x, 0, nmax}], t], x]*Range[0, nmax]!] (* G. C. Greubel, Jun 10 2018 *)

# uses[bell_matrix from A264428]
# Adds a column 1,0,0,0, ... at the left side of the triangle.
bell_matrix(lambda n: (-1)^n if n<2 else 0, 9) # Peter Luschny, Jan 19 2016

E.g.f. : (1 - t)*exp(x*(t - t^2/2)) = 1 + (-1 + x)*t + (-3*x + x^2)*t^2/2! + ... - Peter Bala, Apr 08 2013

A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -2, 3, 1, 0, 10, -5, 6, 1, 0, -80, 30, -5, 10, 1, 0, 880, -290, 45, 5, 15, 1, 0, -12320, 3780, -560, 35, 35, 21, 1, 0, 209440, -61460, 8820, -735, 0, 98, 28, 1, 0, -4188800, 1192800, -167300, 14700, -735, 0, 210, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,      1]
[ 0,      1,      1]
[ 0,     -2,      3,      1]
[ 0,     10,     -5,      6,      1]
[ 0,    -80,     30,     -5,     10,      1]
[ 0,    880,   -290,     45,      5,     15,      1]

Peter Luschny, The Bell transform
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.

Cf. A007696, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265605.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_4_1 = lambda n: prod(4*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_4_1, 8))

A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

Original entry on oeis.org

1, 1, 1, 3, 6, 1, 15, 45, 15, 1, 105, 420, 210, 28, 1, 945, 4725, 3150, 630, 45, 1, 10395, 62370, 51975, 13860, 1485, 66, 1, 135135, 945945, 945945, 315315, 45045, 3003, 91, 1, 2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1, 34459425

Offset: 0

Paul Barry, Apr 12 2010

Row sums are A066223. Reverse of A119743. Inverse is alternating sign version.
Diagonal sums are essentially A025164.
From Tom Copeland, Dec 13 2015: (Start)
See A099174 for relations to the Hermite polynomials and the link for operator relations, including the infinitesimal generator containing A000384.
Row polynomials are 2^n n! Lag(n,-x/2,-1/2), where Lag(n,x,q) is the associated Laguerre polynomial of order q.
The triangles of Bessel numbers entries A122848, A049403, A096713, A104556 contain these polynomials as even or odd rows. Also the aerated version A099174 and A066325. Reversed, these entries are A100861, A144299, A111924.
Divided along the diagonals by the initial element (A001147) of the diagonal, this matrix becomes the even rows of A034839.
(End)
The first few rows appear in expansions related to the Dedekind eta function on pp. 537-538 of the Chan et al. link. - Tom Copeland, Dec 14 2016

			Triangle begins
        1,
        1,        1,
        3,        6,        1,
       15,       45,       15,       1,
      105,      420,      210,      28,       1,
      945,     4725,     3150,     630,      45,      1,
    10395,    62370,    51975,   13860,    1485,     66,    1,
   135135,   945945,   945945,  315315,   45045,   3003,   91,   1,
  2027025, 16216200, 18918900, 7567560, 1351350, 120120, 5460, 120, 1
Production matrix is
  1,  1,
  2,  5,  1,
  0, 12,  9,  1,
  0,  0, 30, 13,  1,
  0,  0,  0, 56, 17,   1,
  0,  0,  0,  0, 90,  21,   1,
  0,  0,  0,  0,  0, 132,  25,   1,
  0,  0,  0,  0,  0,   0, 182,  29,  1,
  0,  0,  0,  0,  0,   0,   0, 240, 33, 1.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Peter Bala, Generalized Dobinski formulas
H. Chan, S. Cooper, and P. Toh, The 26th power of Dedekind's eta function Advances in Mathematics, 207 (2006) 532-543.
Tom Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras, 2012.
Tom Copeland, Juggling Zeros in the Matrix (Example II), 2020.

Cf. A113278.

Cf. A000384, A001147, A034839, A049403, A066325, A096713, A099174, A100861, A104556, A111924, A122848, A144299.

ser := n -> series(KummerU(-n, 1/2, x), x, n+1):
seq(seq((-2)^(n-k)*coeff(ser(n), x, k), k=0..n), n=0..8); # Peter Luschny, Jan 18 2020

t[n_, k_] := k!*Binomial[n, k]/((2 k - n)!*2^(n - k)); u[n_, k_] := t[2 n, k + n]; Table[ u[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jan 14 2011 *)

Number triangle T(n,k) = (2n)!/((2k)!(n-k)!2^(n-k)).
T(n,k) = A122848(2n,k+n). - R. J. Mathar, Jan 14 2011
[x^(1/2)(1+2D)]^2 p(n,x)= p(n+1,x) and [D/(1+2D)]p(n,x)= n p(n-1,x) for the row polynomials of T, with D=d/dx. - Tom Copeland, Dec 26 2012
E.g.f.: exp[t*x/(1-2x)]/(1-2x)^(1/2). - Tom Copeland , Dec 10 2013
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} (2*k+1)*(2*k+3)*...*(2*k+1+2*(n-1))*(x/2)^k/k!. Cf. A113278. - Peter Bala, Jun 23 2014
The raising operator in my 2012 formula expanded is R = [x^(1/2)(1+2D)]^2 = 1 + x + (2 + 4x) D + 4x D^2, which in matrix form acting on an o.g.f. (formal power series) is the transpose of the production array below. The linear term x is the diagonal of ones after transposition. The main diagonal comes from (1 + 4xD) x^n = (1 + 4n) x^n. The last diagonal comes from (2 D + 4 x D^2) x^n = (2 + 4 xD) D x^n = n * (2 + 4(n-1)) x^(n-1). - Tom Copeland, Dec 13 2015
T(n, k) = (-2)^(n-k)*[x^k] KummerU(-n, 1/2, x). - Peter Luschny, Jan 18 2020

A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, -1, 3, 1, 0, 3, -1, 6, 1, 0, -15, 5, 5, 10, 1, 0, 105, -35, 0, 25, 15, 1, 0, -945, 315, -35, 0, 70, 21, 1, 0, 10395, -3465, 490, -35, 70, 154, 28, 1, 0, -135135, 45045, -6895, 630, -105, 378, 294, 36, 1

Offset: 0

Peter Luschny, Dec 30 2015

			[ 1]
[ 0,    1]
[ 0,    1,    1]
[ 0,   -1,    3,    1]
[ 0,    3,   -1,    6,    1]
[ 0,  -15,    5,    5,   10,    1]
[ 0,  105,  -35,    0,   25,   15,    1]
[ 0, -945,  315,  -35,    0,   70,   21,    1]

Peter Luschny, The Bell transform
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.

Cf. A007559, A264428, A264429.

Inverse Bell transforms of other multifactorials are: A048993, A049404, A049410, A075497, A075499, A075498, A119275, A122848, A265604.

# uses[bell_transform from A264428]
def inverse_bell_matrix(generator, dim):
    G = [generator(k) for k in srange(dim)]
    row = lambda n: bell_transform(n, G)
    M = matrix(ZZ, [row(n)+[0]*(dim-n-1) for n in srange(dim)]).inverse()
    return matrix(ZZ, dim, lambda n,k: (-1)^(n-k)*M[n,k])
multifact_3_1 = lambda n: prod(3*k + 1 for k in (0..n-1))
print(inverse_bell_matrix(multifact_3_1, 8))

A144644 Triangle in A144643 read by columns downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 0, 15, 25, 10, 1, 0, 0, 25, 90, 65, 15, 1, 0, 0, 35, 280, 350, 140, 21, 1, 0, 0, 35, 770, 1645, 1050, 266, 28, 1, 0, 0, 0, 1855, 6930, 6825, 2646, 462, 36, 1, 0, 0, 0, 3675, 26425, 39795, 22575, 5880, 750, 45, 1

Offset: 0

David Applegate and N. J. A. Sloane, Jan 25 2009

The Bell transform of the sequence "g(n) = 1 if n<4 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,    1;
  0, 1,  7,    6,     1;
  0, 0, 15,   25,    10,      1;
  0, 0, 25,   90,    65,     15,      1;
  0, 0, 35,  280,   350,    140,     21,     1;
  0, 0, 35,  770,  1645,   1050,    266,    28,     1;
  0, 0,  0, 1855,  6930,   6825,   2646,   462,    36,    1;
  0, 0,  0, 3675, 26425,  39795,  22575,  5880,   750,   45,  1;
  0, 0,  0, 5775, 90475, 211750, 172095, 63525, 11880, 1155, 55, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.

Cf. A001681 (row sums), A111246, A122848, A144643, A144645, A151509, A151511, A264428.

function t(n,k)
  if k eq n then return 1;
  elif k le n-1 or n le 0 then return 0;
  else return (&+[Binomial(k-1,j)*t(n-1,k-j-1): j in [0..3]]);
  end if;
end function;
A144644:= func< n,k | t(k,n) >;
[A144644(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 11 2023

With[{r=15}, Table[BellY[n, k, {1,1,1,1}], {n,0,r}, {k,0,n}]]//Flatten (* Jan Mangaldan, May 22 2016 *)

\\ Function bell_matrix is defined in A264428.
B = bell_matrix( n -> {if(n < 4, 1, 0)}, 9); for(n = 0, 9, printp(); for(k = 1, n, print1(B[n,k], " "))); \\ Peter Luschny, Apr 17 2019

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<4 else 0, 12) # Peter Luschny, Jan 19 2016

Bivariate e.g.f. A144644(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G4(x)), where G4(x) = Sum_{i=1..4} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019
From G. C. Greubel, Oct 11 2023: (Start)
T(n, k) = A144643(k, n).
T(n, k) = A144645(n, n-k).
T(n, k) = t(k, n), where t(n, k) = Sum_{j=0..3} binomial(k-1, j) * t(n-1, k-j-1), with t(n,n) = 1, t(n,k) = 0 if n < 1 or n > k.
Sum_{k=0..n} T(n, k) = A001681(n). (End)

A151511 The triangle in A151359 read by rows downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 0, 63, 301, 350, 140, 21, 1, 0, 0, 119, 966, 1701, 1050, 266, 28, 1, 0, 0, 210, 2989, 7770, 6951, 2646, 462, 36, 1, 0, 0, 336, 8925, 33985, 42525, 22827, 5880, 750, 45, 1, 0, 0, 462, 25641

Offset: 0

N. J. A. Sloane, May 14 2009

The Bell transform of g(n) = 1 if n<6 else 0. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
1
0 1
0 1 1
0 1 3 1
0 1 7 6 1
0 1 15 25 10 1
0 1 31 90 65 15 1
0 0 63 301 350 140 21 1
0 0 119 966 1701 1050 266 28 1

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394, 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009 (see Table 7 E5(n,k) page 16).

Begins in same way as triangle of Stirling numbers of second kind, A048993, but is strictly different. N. J. A. Sloane, Aug 09 2017

Cf. A148092 (row sums), A122848, A111246, A144644, A151509.

Unprotect[Power]; 0^0 = 1; a[n_ /; 1 <= n <= 6] = 1; a[] = 0; T[n, k_] := T[n, k] = If[k == 0, a[0]^n, Sum[Binomial[n - 1, j - 1] a[j] T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 20 2016, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<6 else 0, 12) # Peter Luschny, Jan 19 2016

Bivariate e.g.f. A151511(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G6(x)), where G6(x) = Sum_{i=1..6} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Row 9 added by Michel Marcus, Feb 13 2014

More rows from R. J. Mathar, May 28 2019

A113278 Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.

Original entry on oeis.org

1, 1, 1, -1, 2, 1, 3, -3, 3, 1, -15, 12, -6, 4, 1, 105, -75, 30, -10, 5, 1, -945, 630, -225, 60, -15, 6, 1, 10395, -6615, 2205, -525, 105, -21, 7, 1, -135135, 83160, -26460, 5880, -1050, 168, -28, 8, 1, 2027025, -1216215, 374220, -79380, 13230, -1890, 252, -36, 9, 1

Offset: 0

Paul D. Hanna, Oct 22 2005

			Triangle begins:
  1;
  1,1;
  -1,2,1;
  3,-3,3,1;
  -15,12,-6,4,1;
  105,-75,30,-10,5,1;
  -945,630,-225,60,-15,6,1;
  10395,-6615,2205,-525,105,-21,7,1;
  ...
where T(n,k) = (-1)^(n-1-k)*A001147(n-1)*C(n,k).
The matrix square equals:
  1;
  2,1;
  0,4,1;
  0,0,6,1;
  0,0,0,8,1;
  0,0,0,0,10,1;
  0,0,0,0,0,12,1;
  ...
The matrix log, L, begins:
  0;
  1,0;
  -2,2,0;
  8,-6,3,0;
  -48,32,-12,4,0;
  384,-240,80,-20,5,0;
  -3840,2304,-720,160,-30,6,0;
  ...
where L(n,k) = (-1)^(n-1-k)*A000165(n-1)*C(n,k).

Peter Bala, Generalized Dobinski formulas

Cf. A039683, A122848.

Cf. A001147 (odd double factorials), A000165 (even double factorials).

(* The function RiordanArray is defined in A256893. *)
RiordanArray[Sqrt[1 + 2 #]&, #&, 10, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r==c,1,if(r==c+1,2*c)))); (sum(i=0,n+1,(sum(j=1,n+1,-(M^0-M)^j/j)/2)^i/i!))[n+1,k+1]}

Exponential Riordan array [sqrt(1 + 2*x),x] with e.g.f. sqrt(1+2*x)*exp(t*x) = 1 + (1+t)*x + (-1+2*t+t^2)*x^2/2! + ... . 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..inf} (2*k+1)*(2*k-1)*...*(2*k+1-2*(n-1))*(x/2)^k/k!. Cf. A122848. - Peter Bala, Jun 23 2014

A151509 The triangle in A151338 read by rows downwards.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 0, 31, 90, 65, 15, 1, 0, 0, 56, 301, 350, 140, 21, 1, 0, 0, 91, 938, 1701, 1050, 266, 28, 1, 0, 0, 126, 2737, 7686, 6951, 2646, 462, 36, 1, 0, 0, 126, 7455, 32725, 42315, 22827, 5880, 750, 45, 1, 0, 0, 0, 18711, 132055

Offset: 0

N. J. A. Sloane, May 14 2009

The Bell transform of the sequence "g(n) = 1 if n < 5, otherwise 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  3,   1;
  0, 1,  7,   6,    1;
  0, 1, 15,  25,   10,    1;
  0, 0, 31,  90,   65,   15,   1;
  0, 0, 56, 301,  350,  140,  21,  1;
  0, 0, 91, 938, 1701, 1050, 266, 28, 1;

Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009 (see Table 6 E4(n,k) page 15).

Cf. A110038 (row sums), A122848, A111246, A144644, A151511.

rows = 10;
BellMatrix[f_Function | f_Symbol, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[If[# < 5, 1, 0]&, rows];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<5 else 0, 12) # Peter Luschny, Jan 19 2016

Bivariate e.g.f A151509(x,t) = Sum_{n>=0, k>=0} T(n,k)*x^n*t^k/n! = exp(t*G5(x)), where G5(x) = Sum_{i=1..5} x^i/i! is the e.g.f. of column 1. - R. J. Mathar, May 28 2019

Row 9 added by Michel Marcus, Feb 13 2014

More rows from R. J. Mathar, May 28 2019

A122850 Exponential Riordan array (1, sqrt(1+2x)-1).

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 3, -3, 1, 0, -15, 15, -6, 1, 0, 105, -105, 45, -10, 1, 0, -945, 945, -420, 105, -15, 1, 0, 10395, -10395, 4725, -1260, 210, -21, 1, 0, -135135, 135135, -62370, 17325, -3150, 378, -28, 1, 0, 2027025, -2027025, 945945, -270270, 51975, -6930, 630, -36, 1

Offset: 0

Paul Barry, Sep 14 2006

Inverse of number triangle A122848. Entries are Bessel polynomial coefficients. Row sums are A000806.

Also the inverse Bell transform of the sequence "g(n) = 1 if n<2 else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 19 2016

			Triangle begins
  1
  0 1
  0 -1 1
  0 3 -3 1
  0 -15 15 -6 1
  0 105 -105 45 -10 1
  0 -945 945 -420 105 -15 1
  0 10395 -10395 4725 -1260 210 -21 1
  0 -135135 135135 -62370 17325 -3150 378 -28 1
  0 2027025 -2027025 945945 -270270 51975 -6930 630 -36 1
  0 -34459425 34459425 -16216200 4729725 -945945 135135 -13860 990 -45 1
  ...

P. Bala, The white diamond product of power series
Orli Herscovici, Study of the p,q-deformed Touchard polynomials, arXiv:1904.07674 [math.CO], 2019.
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Bessel polynomials
S. Willerton, The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials, arXiv:1708.03227v1 [math.MG], 2017.

Cf. A000806, A001497, A008277, A039757, A122848, A132062, A144301.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> (-1)^n*doublefactorial(2*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 = 12;
M = BellMatrix[Function[n, (-1)^n (2n-1)!!], rows];
Table[M[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 26 2018, after Peter Luschny *)

# uses[bell_matrix from A264428]
bell_matrix(lambda n: 1 if n<2 else 0, 12).inverse() # Peter Luschny, Jan 19 2016

T(n,k) = (-1)^(n-k)*A132062(n,k). - Philippe Deléham, Nov 06 2011
Triangle equals the matrix product A039757*A008277. Dobinski-type formula for the row polynomials: R(n,x) = x*exp(-x)*Sum_{k = 0..inf} (k-1)*(k-3)*(k-5)*...*(k-(2*n-3))*x^k/k! for n >= 1. Cf. A001497. - Peter Bala, Jun 23 2014
From Peter Bala, Jan 09 2018: (Start)
Alternative Dobinski-type formula for the row polynomials: R(n,x) = exp(-x)*Sum_{k = 0..inf} k*(k-2)*(k-4)*...*(k-(2*n-2))*x^k/k!.
Equivalently, R(n,x) = x o (x-2) o (x-4) o...o (x-(2*n-2)), where o denotes the white diamond product of polynomials. See the Bala link for the definition and details.
The white diamond products (x-1) o (x-3) o...o (x-(2*n-3)) give the row polynomials of the array with a factor of x removed.
If d is the first derivative operator f -> d/dx(f(x)) and D is the operator f(x) -> 1/x*d/dx(f(x)) then x^(2*n)*D^n = R(n,x*d), with the understanding that (x*d)^k is to interpreted as the operator f(x) -> x^k*d^k(f(x))/dx^k. (End)
Sum_{k=0..n} (-1)^(n+k) * T(n,k) = A144301(n). - Alois P. Heinz, Aug 31 2022

More terms from Alois P. Heinz, Aug 31 2022

A368726 Number of non-isomorphic connected multiset partitions of weight n into singletons or pairs.

Original entry on oeis.org

1, 1, 3, 3, 8, 10, 26, 38, 93, 161, 381, 732, 1721, 3566, 8369, 18316, 43280, 98401, 234959, 549628, 1327726, 3175670, 7763500, 18905703, 46762513, 115613599, 289185492, 724438500, 1831398264, 4641907993, 11853385002, 30365353560

Offset: 0

Gus Wiseman, Jan 06 2024

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 10 multiset partitions:
  {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}      {{1},{1,1},{1,1}}
         {{1,2}}    {{2},{1,2}}    {{1,2},{1,2}}      {{1},{1,2},{2,2}}
         {{1},{1}}  {{1},{1},{1}}  {{1,2},{2,2}}      {{2},{1,2},{1,2}}
                                   {{1,3},{2,3}}      {{2},{1,2},{2,2}}
                                   {{1},{1},{1,1}}    {{2},{1,3},{2,3}}
                                   {{1},{2},{1,2}}    {{3},{1,3},{2,3}}
                                   {{2},{2},{1,2}}    {{1},{1},{1},{1,1}}
                                   {{1},{1},{1},{1}}  {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

For edges of any size we have A007718.
This is the connected case of A320663.
The case of singletons and strict pairs is A368727, Euler transform A339888.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A007716 counts non-isomorphic multiset partitions, into pairs A007717.
A062740 counts connected loop-graphs, unlabeled A054921.
A320732 counts factorizations into primes or semiprimes, strict A339839.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A001515, A000666, A122848, A283877, A302545, A320462, A321405, A368186, A368598, A368599, A368731.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort/@(#/.x_Integer:>s[[x]])]& /@ sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n]}];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}]]];
Table[Length[Union[brute /@ Select[mpm[n], Max@@Length/@#<=2&&Length[csm[#]]<=1&]]],{n,0,8}]

Inverse Euler transform of A320663.

cp's OEIS Frontend

A104556 Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.

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A265604 Triangle read by rows: The inverse Bell transform of the quartic factorial numbers (A007696).

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A176230 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

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Maple

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A265605 Triangle read by rows: The inverse Bell transform of the triple factorial numbers (A007559).

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Sage

A144644 Triangle in A144643 read by columns downwards.

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Magma

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A151511 The triangle in A151359 read by rows downwards.

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A113278 Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where [T^2](n,n) = 1 and [T^2](n+1,n) = 2*(n+1) for n>=0.

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