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A338805
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).
7
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
1,2
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
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
Column k=1..3 give A318249, A338810, A338811.
Row sums give A028342.
Cf. A000005 (d(n)), A008298, A264428.
Sequence in context: A091543 A330858 A059575 * A120769 A187141 A165604
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 10 2020
STATUS
approved