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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.

%I A338865 #28 Nov 14 2020 11:04:12
%S A338865 1,6,1,24,18,1,168,204,36,1,720,2280,780,60,1,8640,25200,14400,2100,
%T A338865 90,1,40320,292320,252000,58800,4620,126,1,604800,3729600,4334400,
%U A338865 1486800,183120,8904,168,1,4717440,46811520,76265280,35743680,6335280,474768,15624,216,1
%N 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.
%H A338865 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A338865 E.g.f.: exp(Sum_{n>0} u*sigma(n)*x^n).
%F A338865 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.
%F A338865 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).
%e A338865 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! + ... .
%e A338865 Triangle begins:
%e A338865        1;
%e A338865        6,       1;
%e A338865       24,      18,       1;
%e A338865      168,     204,      36,       1;
%e A338865      720,    2280,     780,      60,      1;
%e A338865     8640,   25200,   14400,    2100,     90,    1;
%e A338865    40320,  292320,  252000,   58800,   4620,  126,   1;
%e A338865   604800, 3729600, 4334400, 1486800, 183120, 8904, 168, 1;
%e A338865   ...
%t A338865 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 *)
%o A338865 (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)}
%o A338865 (PARI) a(n) = if(n<1, 0, n!*sigma(n));
%o A338865 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 A338865 Column k=1..2 give n! * sigma(n), (n!/2) * A000385(n-1).
%Y A338865 Rows sum give A294361.
%Y A338865 Cf. A000203 (sigma(n)), A008298, A338864, A338871.
%K A338865 nonn,tabl
%O A338865 1,2
%A A338865 _Seiichi Manyama_, Nov 13 2020