A338871 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*sigma(n)*x^n/n!).

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, 13, 10323, 181063, 426006, 253281, 50022, 3738, 108, 1, 18, 28171, 818760, 3128245, 2956065, 937587, 116760, 6150, 135, 1

Offset: 1

Author

Seiichi Manyama, Nov 13 2020

Keywords

• nonn
• tabl

Comments

Also the Bell transform of A000203.

Examples

			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;
  ...
		

Links

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

Crossrefs

Column k=1..2 give A000203, A330088(n-1).

Row sums give A274804.

Cf. A008298, A338865, A338870.

Programs

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 *)

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)))

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)!.