|
|
A320019
|
|
Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.
|
|
2
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
|
|
LINKS
|
|
|
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:
# Uses function PMatrix from A357368.
|
|
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}]]]];
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|