A338865 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(j*x^j/(1 - x^j)) )^u.

1, 6, 1, 24, 18, 1, 168, 204, 36, 1, 720, 2280, 780, 60, 1, 8640, 25200, 14400, 2100, 90, 1, 40320, 292320, 252000, 58800, 4620, 126, 1, 604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1, 4717440, 46811520, 76265280, 35743680, 6335280, 474768, 15624, 216, 1

Offset: 1

Author

Seiichi Manyama, Nov 13 2020

Keywords

• nonn
• tabl

Examples

			exp(Sum_{n>0} u*sigma(n)*x^n) = 1 + u*x + (6*u+u^2)*x^2/2! + (24*u+18*u^2+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       6,       1;
      24,      18,       1;
     168,     204,      36,       1;
     720,    2280,     780,      60,      1;
    8640,   25200,   14400,    2100,     90,    1;
   40320,  292320,  252000,   58800,   4620,  126,   1;
  604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
  ...
		

Links

• Peter Luschny, The Bell transform.

Crossrefs

Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).

Rows sum give A294361.

Cf. A000203 (sigma(n)), A008298, 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] * j! * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 13 2020 *)

PARI

{T(n, k) = my(u='u); n!*polcoef(polcoef(prod(j=1, n, exp(j*x^j/(1-x^j+x*O(x^n)))^u), n), k)}

PARI

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

E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^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} k*sigma(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} sigma(i_j).