A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*d(n)*x^n/n!), where d(n) is the number of divisors of n.

1, 2, 1, 2, 6, 1, 3, 20, 12, 1, 2, 55, 80, 20, 1, 4, 142, 405, 220, 30, 1, 2, 322, 1792, 1785, 490, 42, 1, 4, 779, 7224, 12152, 5810, 952, 56, 1, 3, 1608, 27323, 73920, 56532, 15498, 1680, 72, 1, 4, 3894, 99690, 414815, 482160, 204204, 35910, 2760, 90, 1

Offset: 1

Author

Seiichi Manyama, Nov 13 2020

Keywords

• nonn
• tabl

Comments

Also the Bell transform of A000005.

Examples

			exp(Sum_{n>0} u*d(n)*x^n/n!) = 1 + u*x + (2*u+u^2)*x^2/2! + (2*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
  1;
  2,   1;
  2,   6,    1;
  3,  20,   12,     1;
  2,  55,   80,    20,    1;
  4, 142,  405,   220,   30,   1;
  2, 322, 1792,  1785,  490,  42,  1;
  4, 779, 7224, 12152, 5810, 952, 56, 1;
  ...
		

Links

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

Crossrefs

Column k=1..2 give A000005, A328681(n-1).

Row sums give A295739.

Cf. A338805, A338864, A338871.

Programs

Mathematica

T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[0, 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 *)

PARI

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

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)*d(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} d(i_j)/(i_j)!.