A319083 Coefficients of polynomials related to the D'Arcais polynomials and Dedekind's eta(q) function, triangle read by rows, T(n,k) for 0 <= k <= n.
1, 0, 1, 0, 3, 1, 0, 4, 6, 1, 0, 7, 17, 9, 1, 0, 6, 38, 39, 12, 1, 0, 12, 70, 120, 70, 15, 1, 0, 8, 116, 300, 280, 110, 18, 1, 0, 15, 185, 645, 885, 545, 159, 21, 1, 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1, 0, 18, 384, 2262, 5586, 6713, 4281, 1498, 284, 27, 1
Offset: 0
Examples
Triangle starts: [0] 1; [1] 0, 1; [2] 0, 3, 1; [3] 0, 4, 6, 1; [4] 0, 7, 17, 9, 1; [5] 0, 6, 38, 39, 12, 1; [6] 0, 12, 70, 120, 70, 15, 1; [7] 0, 8, 116, 300, 280, 110, 18, 1; [8] 0, 15, 185, 645, 885, 545, 159, 21, 1; [9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 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:-sigma(n-k)*P(k,x), k=0..n-1) fi end: Trow := n -> seq(coeff(P(n, x), x, k), k=0..n): seq(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[sigma](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..10); # Alois P. Heinz, Feb 01 2021 # Uses function PMatrix from A357368. PMatrix(10, NumberTheory:-sigma); # 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[1, 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, 10}] // 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} sigma(n-k)*p(k, x).
Sum_{k=0..n} (-1)^k * T(n,k) = A283334(n). - Alois P. Heinz, Feb 07 2025
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