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A338805
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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).
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7
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1, 2, 1, 4, 6, 1, 18, 28, 12, 1, 48, 170, 100, 20, 1, 480, 988, 870, 260, 30, 1, 1440, 7896, 7588, 3150, 560, 42, 1, 20160, 60492, 73808, 37408, 9100, 1064, 56, 1, 120960, 555264, 764524, 460656, 140448, 22428, 1848, 72, 1, 1451520, 5819904, 8448120, 5952700, 2162160, 436296, 49140, 3000, 90, 1
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OFFSET
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1,2
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COMMENTS
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Also the Bell transform of A318249.
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LINKS
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FORMULA
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E.g.f.: exp(Sum_{n>0} u*d(n)*x^n/n), where d(n) is the number of divisors of n.
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} d(k)*T(n-k; u)/(n-k)!, 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} d(i_j)/i_j.
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EXAMPLE
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exp(Sum_{n>0} u*d(n)*x^n/n) = 1 + u*x + (2*u+u^2)*x^2/2! + (4*u+6*u^2+u^3)*x^3/3! + ... .
Triangle begins:
1;
2, 1;
4, 6, 1;
18, 28, 12, 1;
48, 170, 100, 20, 1;
480, 988, 870, 260, 30, 1;
1440, 7896, 7588, 3150, 560, 42, 1;
20160, 60492, 73808, 37408, 9100, 1064, 56, 1;
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MAPLE
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# The function BellMatrix is defined in A264428 (with column k = 0).
BellMatrix(n -> n!*NumberTheory:-SumOfDivisors(n+1, 0), 9);
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
(1/n)*x*add(NumberTheory:-SumOfDivisors(n-k, 0)*P(k, x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k = 1..n):
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 0, (n - 1)! * DivisorSigma[0, n]]; 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 *)
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PROG
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(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)}
(PARI) a(n) = if(n<1, 0, (n-1)!*numdiv(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)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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