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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. 5
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Column k is the k-fold self-convolution of sigma (A000203). - Alois P. Heinz, Feb 01 2021
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} sigma(n-k)*p(k, x).
EXAMPLE
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;
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 *)
CROSSREFS
Columns k=0..2 give: A000007, A000203, A000385.
Row sums are A180305.
T(2n,n) gives A340993.
Sequence in context: A274662 A186827 A207327 * A332099 A045406 A143468
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Oct 03 2018
STATUS
approved

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Last modified May 18 02:38 EDT 2024. Contains 372617 sequences. (Running on oeis4.)