cp's OEIS Frontend

A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*d(n)*x^n/n!), where d(n) is the number of divisors of n.

%I A338870 #27 Nov 15 2020 01:53:33
%S A338870 1,2,1,2,6,1,3,20,12,1,2,55,80,20,1,4,142,405,220,30,1,2,322,1792,
%T A338870 1785,490,42,1,4,779,7224,12152,5810,952,56,1,3,1608,27323,73920,
%U A338870 56532,15498,1680,72,1,4,3894,99690,414815,482160,204204,35910,2760,90,1
%N A338870 Triangle T(n,k) defined by Sum_{k=1..n} T(n,k)*u^k*x^n/n! = exp(Sum_{n>0} u*d(n)*x^n/n!), where d(n) is the number of divisors of n.
%C A338870 Also the Bell transform of A000005.
%H A338870 Seiichi Manyama, <a href="/A338870/b338870.txt">Rows n = 1..100, flattened</a>
%H A338870 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/BellTransform">The Bell transform</a>.
%F A338870 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)*d(k)*T(n-k; u), T(0; u) = 1.
%F A338870 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)/(i_j)!.
%e A338870 exp(Sum_{n>0} u*d(n)*x^n/n!) = 1 + u*x + (2*u+u^2)*x^2/2! + (2*u+6*u^2+u^3)*x^3/3! + ... .
%e A338870 Triangle begins:
%e A338870   1;
%e A338870   2,   1;
%e A338870   2,   6,    1;
%e A338870   3,  20,   12,     1;
%e A338870   2,  55,   80,    20,    1;
%e A338870   4, 142,  405,   220,   30,   1;
%e A338870   2, 322, 1792,  1785,  490,  42,  1;
%e A338870   4, 779, 7224, 12152, 5810, 952, 56, 1;
%e A338870   ...
%t A338870 T[n_, 0] := Boole[n == 0]; T[n_, k_] := T[n, k] = Sum[Boole[j > 0] * Binomial[n - 1, j - 1] * DivisorSigma[0, 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 A338870 (PARI) a(n) = if(n<1, 0, numdiv(n));
%o A338870 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 A338870 Column k=1..2 give A000005, A328681(n-1).
%Y A338870 Row sums give A295739.
%Y A338870 Cf. A338805, A338864, A338871.
%K A338870 nonn,tabl
%O A338870 1,2
%A A338870 _Seiichi Manyama_, Nov 13 2020