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A338864 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = Product_{j>0} ( exp(x^j/(1 - x^j)) )^u. 2
1, 4, 1, 12, 12, 1, 72, 96, 24, 1, 240, 840, 360, 40, 1, 2880, 7200, 4920, 960, 60, 1, 10080, 70560, 65520, 19320, 2100, 84, 1, 161280, 745920, 887040, 362880, 58800, 4032, 112, 1, 1088640, 7983360, 12640320, 6652800, 1481760, 150192, 7056, 144, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also the Bell transform of A323295.

LINKS

Table of n, a(n) for n=1..45.

Peter Luschny, The Bell transform.

FORMULA

E.g.f.: exp(Sum_{n>0} u*d(n)*x^n), where d(n) is the number of divisors of 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*d(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} d(i_j).

EXAMPLE

exp(Sum_{n>0} u*d(n)*x^n) = 1 + u*x + (4*u+u^2)*x^2/2! + (12*u+12*u^2+u^3)*x^3/3! + ... .

Triangle begins:

       1;

       4,      1;

      12,     12,      1;

      72,     96,     24,      1;

     240,    840,    360,     40,     1;

    2880,   7200,   4920,    960,    60,    1;

   10080,  70560,  65520,  19320,  2100,   84,   1;

  161280, 745920, 887040, 362880, 58800, 4032, 112, 1;

  ...

MATHEMATICA

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

PROG

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

(PARI) a(n) = if(n<1, 0, n!*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 A323295, (n!/2) * A055507(n-1).

Rows sum give A294363.

Cf. A000005 (d(n)), A338805, A338865, A338870.

Sequence in context: A144878 A049424 A157394 * A078219 A187541 A117413

Adjacent sequences:  A338861 A338862 A338863 * A338865 A338866 A338867

KEYWORD

nonn,tabl

AUTHOR

Seiichi Manyama, Nov 13 2020

STATUS

approved

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Last modified April 20 12:39 EDT 2021. Contains 343135 sequences. (Running on oeis4.)