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

0,5

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

Columns k=0-3 give: A000007, A000005, A055507, A191829.

Row sums are A129921.

T(2n,n) gives A340992.

Cf. A319083.

Sequence in context: A122542 A227341 A098542 * A141343 A256678 A066709

Adjacent sequences: A320016 A320017 A320018 * A320020 A320021 A320022

Keyword

nonn,tabl

Author

Peter Luschny, Oct 03 2018

Status

approved