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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!). 3
 1, 3, 1, 4, 9, 1, 7, 43, 18, 1, 6, 155, 175, 30, 1, 12, 511, 1230, 485, 45, 1, 8, 1442, 7231, 5600, 1085, 63, 1, 15, 4131, 37870, 52381, 18550, 2114, 84, 1, 13, 10323, 181063, 426006, 253281, 50022, 3738, 108, 1, 18, 28171, 818760, 3128245, 2956065, 937587, 116760, 6150, 135, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also the Bell transform of A000203. LINKS Seiichi Manyama, Rows n = 1..100, flattened Peter Luschny, The Bell transform. FORMULA 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. 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)!. EXAMPLE 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! + ... . Triangle begins: 1; 3, 1; 4, 9, 1; 7, 43, 18, 1; 6, 155, 175, 30, 1; 12, 511, 1230, 485, 45, 1; 8, 1442, 7231, 5600, 1085, 63, 1; 15, 4131, 37870, 52381, 18550, 2114, 84, 1; ... MATHEMATICA 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 *) PROG (PARI) a(n) = if(n<1, 0, sigma(n)); 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))) CROSSREFS Column k=1..2 give A000203, A330088(n-1). Row sums give A274804. Cf. A008298, A338865, A338870. Sequence in context: A220605 A094166 A266131 * A202353 A108621 A193792 Adjacent sequences: A338868 A338869 A338870 * A338872 A338873 A338874 KEYWORD nonn,tabl AUTHOR Seiichi Manyama, Nov 13 2020 STATUS approved

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Last modified August 7 03:39 EDT 2024. Contains 375008 sequences. (Running on oeis4.)