%I #35 Jun 01 2022 10:18:14
%S 1,2,1,4,6,1,18,28,12,1,48,170,100,20,1,480,988,870,260,30,1,1440,
%T 7896,7588,3150,560,42,1,20160,60492,73808,37408,9100,1064,56,1,
%U 120960,555264,764524,460656,140448,22428,1848,72,1,1451520,5819904,8448120,5952700,2162160,436296,49140,3000,90,1
%N Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1-x^j)^(-u/j).
%C Also the Bell transform of A318249.
%C 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
%H Seiichi Manyama, <a href="/A338805/b338805.txt">Rows n = 1..100, flattened</a>
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F E.g.f.: exp(Sum_{n>0} u*d(n)*x^n/n), where d(n) is the number of divisors of n.
%F 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.
%F 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.
%e 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! + ... .
%e Triangle begins:
%e 1;
%e 2, 1;
%e 4, 6, 1;
%e 18, 28, 12, 1;
%e 48, 170, 100, 20, 1;
%e 480, 988, 870, 260, 30, 1;
%e 1440, 7896, 7588, 3150, 560, 42, 1;
%e 20160, 60492, 73808, 37408, 9100, 1064, 56, 1;
%p # The function BellMatrix is defined in A264428 (with column k = 0).
%p BellMatrix(n -> n!*NumberTheory:-SumOfDivisors(n+1, 0), 9);
%p # Alternative:
%p P := proc(n, x) option remember; if n = 0 then 1 else
%p (1/n)*x*add(NumberTheory:-SumOfDivisors(n-k, 0)*P(k, x), k=0..n-1) fi end:
%p Trow := n -> seq(n!*coeff(P(n, x), x, k), k = 1..n):
%p seq(Trow(n), n = 0..10); # _Peter Luschny_, Jun 01 2022
%t 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 *)
%o (PARI) {T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, (1-x^j+x*O(x^n))^(-u/j)), n), k)}
%o (PARI) a(n) = if(n<1, 0, (n-1)!*numdiv(n));
%o 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)))
%Y Column k=1..3 give A318249, A338810, A338811.
%Y Row sums give A028342.
%Y Cf. A000005 (d(n)), A008298, A264428.
%K nonn,tabl
%O 1,2
%A _Seiichi Manyama_, Nov 10 2020