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%I #29 Nov 14 2020 11:04:05
%S 1,4,1,12,12,1,72,96,24,1,240,840,360,40,1,2880,7200,4920,960,60,1,
%T 10080,70560,65520,19320,2100,84,1,161280,745920,887040,362880,58800,
%U 4032,112,1,1088640,7983360,12640320,6652800,1481760,150192,7056,144,1
%N 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.
%C Also the Bell transform of A323295.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F E.g.f.: exp(Sum_{n>0} u*d(n)*x^n), where d(n) is the number of divisors of n.
%F 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.
%F 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).
%e 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! + ... .
%e Triangle begins:
%e 1;
%e 4, 1;
%e 12, 12, 1;
%e 72, 96, 24, 1;
%e 240, 840, 360, 40, 1;
%e 2880, 7200, 4920, 960, 60, 1;
%e 10080, 70560, 65520, 19320, 2100, 84, 1;
%e 161280, 745920, 887040, 362880, 58800, 4032, 112, 1;
%e ...
%t 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 *)
%o (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)}
%o (PARI) a(n) = if(n<1, 0, n!*numdiv(n));
%o 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)))
%Y Column k=1..2 give A323295, (n!/2) * A055507(n-1).
%Y Rows sum give A294363.
%Y Cf. A000005 (d(n)), A338805, A338865, A338870.
%K nonn,tabl
%O 1,2
%A _Seiichi Manyama_, Nov 13 2020
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