OFFSET
1,2
COMMENTS
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
LINKS
Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.
FORMULA
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.
EXAMPLE
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;
MAPLE
# 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
MATHEMATICA
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 *)
PROG
(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)}
(PARI) 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)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 10 2020
STATUS
approved