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A338871 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*sigma(n)*x^n/n!).

%I A338871 #25 Nov 15 2020 01:53:40
%S A338871 1,3,1,4,9,1,7,43,18,1,6,155,175,30,1,12,511,1230,485,45,1,8,1442,
%T A338871 7231,5600,1085,63,1,15,4131,37870,52381,18550,2114,84,1,13,10323,
%U A338871 181063,426006,253281,50022,3738,108,1,18,28171,818760,3128245,2956065,937587,116760,6150,135,1
%N A338871 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*sigma(n)*x^n/n!).
%C A338871 Also the Bell transform of A000203.
%H A338871 Seiichi Manyama, <a href="/A338871/b338871.txt">Rows n = 1..100, flattened</a>
%H A338871 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A338871 T(n; u) = Sum_{k=1..n} T(n,k)*u^k is given by T(n; u) = u * Sum_{k=1..n} binomial(n-1,k-1)*sigma(k)*T(n-k; u), T(0; u) = 1.
%F A338871 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)/(i_j)!.
%e A338871 exp(Sum_{n>0} u*sigma(n)*x^n/n!) = 1 + u*x + (3*u+u^2)*x^2/2! + (4*u+9*u^2+u^3)*x^3/3! + ... .
%e A338871 Triangle begins:
%e A338871    1;
%e A338871    3,    1;
%e A338871    4,    9,     1;
%e A338871    7,   43,    18,     1;
%e A338871    6,  155,   175,    30,     1;
%e A338871   12,  511,  1230,   485,    45,    1;
%e A338871    8, 1442,  7231,  5600,  1085,   63,  1;
%e A338871   15, 4131, 37870, 52381, 18550, 2114, 84, 1;
%e A338871   ...
%t A338871 T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[1, j] * T[n - j, k - 1], {j, 0, n - k + 1}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Amiram Eldar_, Nov 13 2020 *)
%o A338871 (PARI) a(n) = if(n<1, 0, sigma(n));
%o A338871 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 A338871 Column k=1..2 give A000203, A330088(n-1).
%Y A338871 Row sums give A274804.
%Y A338871 Cf. A008298, A338865, A338870.
%K A338871 nonn,tabl
%O A338871 1,2
%A A338871 _Seiichi Manyama_, Nov 13 2020