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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).

%I A338813 #20 Apr 28 2021 02:03:43
%S A338813 1,0,1,4,0,1,-6,16,0,1,48,-30,40,0,1,0,448,-90,80,0,1,1440,-840,2128,
%T A338813 -210,140,0,1,-10080,23532,-6720,7168,-420,224,0,1,120960,-127008,
%U A338813 177868,-30240,19488,-756,336,0,1,0,2191104,-1018080,892540,-100800,45696,-1260,480,0,1
%N 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).
%C A338813 Also the Bell transform of A338814.
%H A338813 Seiichi Manyama, <a href="/A338813/b338813.txt">Rows n = 1..100, flattened</a>
%H A338813 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A338813 E.g.f.: exp(Sum_{n>0} u*A048272(n)*x^n/n).
%F A338813 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.
%F A338813 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.
%e A338813 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! + ... .
%e A338813 Triangle begins:
%e A338813        1;
%e A338813        0,     1;
%e A338813        4,     0,     1;
%e A338813       -6,    16,     0,    1;
%e A338813       48,   -30,    40,    0,    1;
%e A338813        0,   448,   -90,   80,    0,   1;
%e A338813     1440,  -840,  2128, -210,  140,   0, 1;
%e A338813   -10080, 23532, -6720, 7168, -420, 224, 0, 1;
%e A338813   ...
%t A338813 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 *)
%o A338813 (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)}
%o A338813 (PARI) a(n) = if(n<1, 0, (n-1)!*sumdiv(n, d, (-1)^(d+1)));
%o A338813 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 A338813 Column k=1 gives A338814.
%Y A338813 Row sums give A168243.
%Y A338813 Cf. A048272, A075525, A338805.
%K A338813 sign,tabl
%O A338813 1,4
%A A338813 _Seiichi Manyama_, Nov 10 2020