OFFSET
1,2
COMMENTS
Also the Bell transform of A000203.
LINKS
Seiichi Manyama, Rows n = 1..100, flattened
Peter Luschny, The Bell transform.
FORMULA
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)!.
EXAMPLE
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;
...
MATHEMATICA
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 *)
PROG
(PARI) 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)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 13 2020
STATUS
approved