A008298 -id:A008298 - cp's OEIS frontend

A059356 A diagonal of triangle in A008298.

1, 9, 59, 450, 3394, 30912, 293292, 3032208, 36290736, 433762560, 5925016800, 83648747520, 1335385128960, 20323375994880, 376785057196800, 6493118120294400, 132672192555571200, 2513351450024755200, 56577426980420505600, 1188283280226545664000, 29682641812682686464000, 658094690655791972352000

Offset: 2

N. J. A. Sloane, Jan 27 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

Seiichi Manyama, Table of n, a(n) for n = 2..448

Cf. A000203, A000385, A008298.

nmax = 30; Table[n!/2 * Sum[DivisorSigma[1, k] * DivisorSigma[1, n-k] / k / (n-k), {k, 1, n-1}], {n, 2, nmax}] (* Vaclav Kotesovec, Nov 09 2020 *)

{a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-t)), n), 2)} \\ Seiichi Manyama, Nov 07 2020

{a(n)= (n-1)!*sum(k=1, n-1, sigma(k)*sigma(n-k)/k)} \\ Seiichi Manyama, Nov 09 2020

{a(n)= n!*sum(k=1, n-1, sigma(k)*sigma(n-k)/(k*(n-k)))/2} \\ Seiichi Manyama, Nov 09 2020

a(n) = (n-1)! * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/k = (n!/2) * Sum_{k=1..n-1} sigma(k)*sigma(n-k)/(k*(n-k)). - Seiichi Manyama, Nov 09 2020.

E.g.f.: (1/2) * log( Product_{k>=1} (1 - x^k) )^2. - Ilya Gutkovskiy, Apr 24 2021

More terms from Vladeta Jovovic, Dec 28 2001

A059357 A diagonal of triangle in A008298.

Original entry on oeis.org

1, 18, 215, 2475, 28294, 340116, 4335596, 57773700, 831170736, 12532005288, 201002619168, 3401283910752, 60929911689984, 1143429812726400, 22572470529457920, 468013463441475840, 10124124979606179840

Offset: 3

N. J. A. Sloane, Jan 27 2001

nonn

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159.

Seiichi Manyama, Table of n, a(n) for n = 3..448

Cf. A008298.

nmax = 20; Table[n!/6 * Sum[Sum[Sum[If[i + j + k == n, DivisorSigma[1,i] * DivisorSigma[1,j] * DivisorSigma[1,k] / (i*j*k), 0], {k, 1, n}], {j, 1, n}], {i, 1, n}], {n, 3, nmax}] (* Vaclav Kotesovec, Nov 09 2020 *)

{a(n) = my(t='t); n!*polcoef(polcoef(prod(k=1, n, (1-x^k+x*O(x^n))^(-t)), n), 3)} \\ Seiichi Manyama, Nov 07 2020

a(n) = (n!/6) * Sum_{i,j,k > 0 and i+j+k=n} sigma(i)*sigma(j)*sigma(k)/(i*j*k). - Seiichi Manyama, Nov 09 2020.

E.g.f.: -(1/6) * log( Product_{k>=1} (1 - x^k) )^3. - Ilya Gutkovskiy, Apr 24 2021

More terms from Vladeta Jovovic, Dec 28 2001

A078521 Signed triangle of D'Arcais numbers (A008298) : coefficients of r in the polynomials generated by the series coefficients of z^n in Product[(1-z^k)^r, {k,1,Inf}]*(n!).

Original entry on oeis.org

1, 0, -1, 0, -3, 1, 0, -8, 9, -1, 0, -42, 59, -18, 1, 0, -144, 450, -215, 30, -1, 0, -1440, 3394, -2475, 565, -45, 1, 0, -5760, 30912, -28294, 9345, -1225, 63, -1, 0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1, 0, -524160, 3032208, -4335596, 2341332, -579369, 69552, -4074, 108, -1, 0, -6531840

Offset: 1

Wouter Meeussen, Jan 07 2003

Also the Bell transform of -A038048(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			The z-expansion of Product[(1-z^k)^r, {k,1,3}] is 1 - r*z + ((-3+r)*r*z^2)/2 -(r*(8-9*r +r^2)*z^3)/6, so the third row of the triangle is 0,-8,9,-1.
Triangle begins
1,
0, -1,
0, -3, 1,
0, -8, 9, -1,
0, -42, 59, -18, 1,
0, -144, 450, -215, 30, -1,
0, -1440, 3394, -2475, 565, -45, 1,
0, -5760, 30912, -28294, 9345, -1225, 63, -1,
0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1
...

Cf. A008298, A010815, A038048.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> -n!*numtheory:-sigma(n+1), 9); # Peter Luschny, Jan 26 2016
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
-(1/n)*x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9); # Peter Luschny, Oct 03 2018

w=16;(CoefficientList[ #, r]&/@ CoefficientList[Series[Product[(1-z^k)^r, {k, 1, w}], {z, 0, w}], z])Range[0, w]!
(* 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!*DivisorSigma[1, n + 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

See Mathematica line.

Row sums give A010815 * n!.

A038048 a(n) = (n-1)! * sigma(n).

Original entry on oeis.org

1, 3, 8, 42, 144, 1440, 5760, 75600, 524160, 6531840, 43545600, 1117670400, 6706022400, 149448499200, 2092278988800, 40537905408000, 376610217984000, 13871809695744000, 128047474114560000, 5109094217170944000

Offset: 1

Christian G. Bower

sigma(n) = A000203(n) is the sum of the divisors of n.
Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).
Left edge of triangle in A008298.

			a(6) = 5! * (1 + 2 + 3 + 6) = 1440 = 6! * (1 + 1/2 + 1/3 + 1/6).

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).

T. D. Noe, Table of n, a(n) for n = 1..100
Xiaojun Liu, Motohico Mulase, Adam Sorkin, Quantum curves for simple Hurwitz numbers of an arbitrary base curve, arXiv:1304.0015 [math.AG], 2013.
H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.

Cf. A000203, A057625, A058892, A110373, A110374.

a := n -> n!*add(1/j, j=numtheory:-divisors(n)): seq(a(n), n=1..23); # Emeric Deutsch, Jul 24 2005

a[n_] := (n-1)!*DivisorSigma[1, n]; Table[a[n], {n, 20}] (* Jean-François Alcover, Mar 23 2011 *)

a(n)=(n-1)!*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

A038048 = lambda n: factorial(n-1)*sigma(n,1)
[A038048(n) for n in (1..20)] # Peter Luschny, Jan 19 2016

a(n) = Sum_{d|n} n!/d. - Amarnath Murthy, Jul 24 2005
a(p) = (p+1)*(p-1)! if p is a prime. - Amarnath Murthy, Jul 24 2005
E.g.f.: log(f(x)), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1 - x^k). - N. J. A. Sloane
E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - Vladeta Jovovic, Mar 27 2005
a(n) = A000142(n-1)*A000203(n). - Omar E. Pol, Feb 26 2014

More terms from Emeric Deutsch, Jul 24 2005

Edited by N. J. A. Sloane, May 12 2008 at the suggestion of Joerg Arndt

A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1

Offset: 0

Peter Luschny, Oct 03 2018

Column k is the k-fold self-convolution of sigma (A000203). - Alois P. Heinz, Feb 01 2021

For fixed k, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - Vaclav Kotesovec, Sep 20 2024

			Triangle starts:
[0] 1;
[1] 0,  1;
[2] 0,  3,   1;
[3] 0,  4,   6,    1;
[4] 0,  7,  17,    9,    1;
[5] 0,  6,  38,   39,   12,    1;
[6] 0, 12,  70,  120,   70,   15,   1;
[7] 0,  8, 116,  300,  280,  110,  18,   1;
[8] 0, 15, 185,  645,  885,  545, 159,  21,  1;
[9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Columns k=0..6 give: A000007, A000203, A000385, A374951, A374977, A374978, A374979.
Row sums are A180305.
T(2n,n) gives A340993.
Cf. A008298, A078521, A283334, A319933.

P := proc(n, x) option remember; if n = 0 then 1 else
x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9);
# second Maple program:
T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
      `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
       add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Feb 01 2021
# Uses function PMatrix from A357368.
PMatrix(10, NumberTheory:-sigma); # Peter Luschny, Oct 19 2022

T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
     If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
     With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)

The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} sigma(n-k)*p(k, x).
Sum_{k=0..n} (-1)^k * T(n,k) = A283334(n). - Alois P. Heinz, Feb 07 2025

A075525 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kt^n/n! = ((1+t)(1+t^2)(1+t^3)...)^u.

Original entry on oeis.org

1, 1, 1, 8, 3, 1, 6, 35, 6, 1, 144, 110, 95, 10, 1, 480, 1594, 585, 205, 15, 1, 5760, 8064, 8974, 1995, 385, 21, 1, 5040, 125292, 70252, 35329, 5320, 658, 28, 1, 524160, 684144, 1178540, 392364, 110649, 12096, 1050, 36, 1, 2177280, 14215536, 10683180, 7260560, 1630125, 295113, 24570, 1590, 45, 1

Offset: 1

Vladeta Jovovic, Oct 11 2002

Also the Bell transform of A265024. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

			Triangle begins:
      1;
      1,      1;
      8,      3,     1;
      6,     35,     6,     1;
    144,    110,    95,    10,    1;
    480,   1594,   585,   205,   15,   1;
   5760,   8064,  8974,  1995,  385,  21,  1;
   5040, 125292, 70252, 35329, 5320, 658, 28, 1;
  ...
exp(Sum_{n>0} u*A000593(n)*t^n/n) = 1 + u*t/1! + (u+u^2)*t^2/2! + (8*u+3*u^2+u^3)*t^3/3! + (6*u+35*u^2+6*u^3+u^4)*t^4/4! + ...  - _Seiichi Manyama_, Nov 08 2020.

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..3 give A265024, A338787, A338788.

Cf. A000593, A008298.

# Adds (1,0,0,0,...) as row 0.
seq(PolynomialTools[CoefficientList](n!*coeff(series(mul((1+z^k)^u, k=1..20),z,20),z,n),u), n=0..9); # Peter Luschny, Jan 26 2016

T[n_, k_] := n! SeriesCoefficient[(Times @@ (1 + t^Range[n]))^u, {t, 0, n}, {u, 0, k}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 04 2019 *)

a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)*n/d));
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 (1,0,0,0,..) as row 0.
d = lambda n: sum((-1)^(d+1)*n/d for d in divisors(n))
bell_matrix(lambda n: factorial(n)*d(n+1), 9) # Peter Luschny, Jan 26 2016

Row sums give n!*A000009(n).
From Seiichi Manyama, Nov 08 2020: (Start)
E.g.f.: exp(Sum_{n>0} u*A000593(n)*t^n/n).
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} A000593(k)*T(n-k; u)/(n-k)!, T(0; u) = 1. (End)
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} A000593(i_j)/i_j. - Seiichi Manyama, Nov 09 2020.

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).

Original entry on oeis.org

1, 2, 1, 4, 6, 1, 18, 28, 12, 1, 48, 170, 100, 20, 1, 480, 988, 870, 260, 30, 1, 1440, 7896, 7588, 3150, 560, 42, 1, 20160, 60492, 73808, 37408, 9100, 1064, 56, 1, 120960, 555264, 764524, 460656, 140448, 22428, 1848, 72, 1, 1451520, 5819904, 8448120, 5952700, 2162160, 436296, 49140, 3000, 90, 1

Offset: 1

Seiichi Manyama, Nov 10 2020

Also the Bell transform of A318249.

If we use sigma(n,1) in Vladeta Jovovic's formulas in A008298 then one gets the D'Arcais numbers, if we use sigma(n,0) then this sequence arises. # Peter Luschny, Jun 01 2022

			exp(Sum_{n>0} u*d(n)*x^n/n) = 1 + u*x + (2*u+u^2)*x^2/2! + (4*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
      1;
      2,     1;
      4,     6,     1;
     18,    28,    12,     1;
     48,   170,   100,    20,    1;
    480,   988,   870,   260,   30,    1;
   1440,  7896,  7588,  3150,  560,   42,  1;
  20160, 60492, 73808, 37408, 9100, 1064, 56, 1;

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..3 give A318249, A338810, A338811.
Row sums give A028342.
Cf. A000005 (d(n)), A008298, A264428.

# The function BellMatrix is defined in A264428 (with column k = 0).
BellMatrix(n -> n!*NumberTheory:-SumOfDivisors(n+1, 0), 9);
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
(1/n)*x*add(NumberTheory:-SumOfDivisors(n-k, 0)*P(k, x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k = 1..n):
seq(Trow(n), n = 0..10); # Peter Luschny, Jun 01 2022

a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSigma[0, n]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1-x^j+x*O(x^n))^(-u/j)), n), k)}

a(n) = if(n<1, 0, (n-1)!*numdiv(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)))

E.g.f.: exp(Sum_{n>0} u*d(n)*x^n/n), where d(n) is the number of divisors of n.
T(n; u) = Sum_{k=1..n} T(n, k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} d(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
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} d(i_j)/i_j.

A338871 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = exp(Sum_{n>0} usigma(n)x^n/n!).

Original entry on oeis.org

1, 3, 1, 4, 9, 1, 7, 43, 18, 1, 6, 155, 175, 30, 1, 12, 511, 1230, 485, 45, 1, 8, 1442, 7231, 5600, 1085, 63, 1, 15, 4131, 37870, 52381, 18550, 2114, 84, 1, 13, 10323, 181063, 426006, 253281, 50022, 3738, 108, 1, 18, 28171, 818760, 3128245, 2956065, 937587, 116760, 6150, 135, 1

Offset: 1

Seiichi Manyama, Nov 13 2020

Also the Bell transform of A000203.

			exp(Sum_{n>0} u*sigma(n)*x^n/n!) = 1 + u*x + (3*u+u^2)*x^2/2! + (4*u+9*u^2+u^3)*x^3/3! + ... .
Triangle begins:
   1;
   3,    1;
   4,    9,     1;
   7,   43,    18,     1;
   6,  155,   175,    30,     1;
  12,  511,  1230,   485,    45,    1;
   8, 1442,  7231,  5600,  1085,   63,  1;
  15, 4131, 37870, 52381, 18550, 2114, 84, 1;
  ...

Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.

Column k=1..2 give A000203, A330088(n-1).
Row sums give A274804.
Cf. A008298, A338865, A338870.

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

a(n) = if(n<1, 0, 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)))

T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * Sum_{k=1..n} binomial(n-1,k-1)*sigma(k)*T(n-k; u), T(0; u) = 1.

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)!.

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

Original entry on oeis.org

1, 6, 1, 24, 18, 1, 168, 204, 36, 1, 720, 2280, 780, 60, 1, 8640, 25200, 14400, 2100, 90, 1, 40320, 292320, 252000, 58800, 4620, 126, 1, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1

Offset: 1

Seiichi Manyama, Nov 13 2020

			exp(Sum_{n>0} u*sigma(n)*x^n) = 1 + u*x + (6*u+u^2)*x^2/2! + (24*u+18*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       6,       1;
      24,      18,       1;
     168,     204,      36,       1;
     720,    2280,     780,      60,      1;
    8640,   25200,   14400,    2100,     90,    1;
   40320,  292320,  252000,   58800,   4620,  126,   1;
  604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
  ...

Peter Luschny, The Bell transform.

Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).
Rows sum give A294361.
Cf. A000203 (sigma(n)), A008298, A338864, A338871.

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * j! * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, exp(j*x^j/(1-x^j+x*O(x^n)))^u), n), k)}

a(n) = if(n<1, 0, n!*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)))

E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^n).
T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * (n-1)! * Sum_{k=1..n} k*sigma(k)*T(n-k; u)/(n-k)!, T(0; u) = 1.
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).

cp's OEIS Frontend

A059356 A diagonal of triangle in A008298.

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A059357 A diagonal of triangle in A008298.

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A078521 Signed triangle of D'Arcais numbers (A008298) : coefficients of r in the polynomials generated by the series coefficients of z^n in Product[(1-z^k)^r, {k,1,Inf}]*(n!).

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A038048 a(n) = (n-1)! * sigma(n).

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A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.

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A075525 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*t^n/n! = ((1+t)*(1+t^2)*(1+t^3)...)^u.

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A075525 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kt^n/n! = ((1+t)(1+t^2)(1+t^3)...)^u.

A338805 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} (1-x^j)^(-u/j).

A338871 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = exp(Sum_{n>0} usigma(n)x^n/n!).

A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)u^kx^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.