A338813 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} (1+x^j)^(u/j).

1, 0, 1, 4, 0, 1, -6, 16, 0, 1, 48, -30, 40, 0, 1, 0, 448, -90, 80, 0, 1, 1440, -840, 2128, -210, 140, 0, 1, -10080, 23532, -6720, 7168, -420, 224, 0, 1, 120960, -127008, 177868, -30240, 19488, -756, 336, 0, 1, 0, 2191104, -1018080, 892540, -100800, 45696, -1260, 480, 0, 1

Offset: 1

Author

Seiichi Manyama, Nov 10 2020

Keywords

• sign
• tabl

Comments

Also the Bell transform of A338814.

Examples

			exp(Sum_{n>0} u*A048272(n)*x^n/n) = 1 + u*x + u^2*x^2/2! + (4*u+u^3)*x^3/3! + ... .
Triangle begins:
       1;
       0,     1;
       4,     0,     1;
      -6,    16,     0,    1;
      48,   -30,    40,    0,    1;
       0,   448,   -90,   80,    0,   1;
    1440,  -840,  2128, -210,  140,   0, 1;
  -10080, 23532, -6720, 7168, -420, 224, 0, 1;
  ...
		

Links

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

Crossrefs

Column k=1 gives A338814.

Row sums give A168243.

Cf. A048272, A075525, A338805.

Programs

Mathematica

a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSum[n, (-1)^(# + 1) &]]; T[n_, k_] := T[n, k] = If[k == 0, Boole[n == 0], Sum[a[j] * Binomial[n - 1, j - 1] * T[n - j, k - 1], {j, 0, n - k + 1}]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 28 2021 *)

PARI

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

PARI

a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
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*A048272(n)*x^n/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} A048272(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} A048272(i_j)/i_j.