A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.
1, 0, 1, 0, 2, 1, 0, 2, 4, 1, 0, 3, 8, 6, 1, 0, 2, 14, 18, 8, 1, 0, 4, 20, 41, 32, 10, 1, 0, 2, 28, 78, 92, 50, 12, 1, 0, 4, 37, 132, 216, 175, 72, 14, 1, 0, 3, 44, 209, 440, 490, 298, 98, 16, 1, 0, 4, 58, 306, 814, 1172, 972, 469, 128, 18, 1
Offset: 0
Examples
Triangle starts: [0] 1 [1] 0, 1 [2] 0, 2, 1 [3] 0, 2, 4, 1 [4] 0, 3, 8, 6, 1 [5] 0, 2, 14, 18, 8, 1 [6] 0, 4, 20, 41, 32, 10, 1 [7] 0, 2, 28, 78, 92, 50, 12, 1 [8] 0, 4, 37, 132, 216, 175, 72, 14, 1 [9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
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Maple
P := proc(n, x) option remember; if n = 0 then 1 else x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end: Trow := n -> seq(coeff(P(n, x), x, k), k=0..n): seq(lprint([n], Trow(n)), n=0..9); # second Maple program: T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0), `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q-> add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2)))) end: seq(seq(T(n, k), k=0..n), n=0..12); # Alois P. Heinz, Feb 01 2021 # Uses function PMatrix from A357368. PMatrix(10, NumberTheory:-tau); # Peter Luschny, Oct 19 2022 -
Mathematica
T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0], If[k == 1, If[n == 0, 0, DivisorSigma[0, n]], With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 11 2021, after Alois P. Heinz *)
Formula
The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
Sigma[k](n) computes the sum of the k-th power of positive divisors of n. The recurrence applied with k = 0 gives this triangle, with k = 1 gives A319083.
T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - Alois P. Heinz, Feb 14 2021
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