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

1,2

Comments

Also the Bell transform of A000005.

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)*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)!.

Example

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;
  ...

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 *)

Prog

(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)))

Crossrefs

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

Row sums give A295739.

Cf. A338805, A338864, A338871.

Sequence in context: A307584 A266183 A232483 * A265894 A133644 A265870

Adjacent sequences: A338867 A338868 A338869 * A338871 A338872 A338873

Keyword

nonn,tabl

Author

Seiichi Manyama, Nov 13 2020

Status

approved