OFFSET
0,5
COMMENTS
Column k is the k-fold self-convolution of tau (A000005). - Alois P. Heinz, Feb 01 2021
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
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
EXAMPLE
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
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 03 2018
STATUS
approved