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%I #24 Oct 19 2022 10:45:38
%S 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,
%T 8,116,300,280,110,18,1,0,15,185,645,885,545,159,21,1,0,13,258,1261,
%U 2364,2095,942,217,24,1,0,18,384,2262,5586,6713,4281,1498,284,27,1
%N 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.
%C Column k is the k-fold self-convolution of sigma (A000203). - _Alois P. Heinz_, Feb 01 2021
%H Alois P. Heinz, <a href="/A319083/b319083.txt">Rows n = 0..200, flattened</a>
%F The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
%F p(n, x) = x*Sum_{k=0..n-1} sigma(n-k)*p(k, x).
%e Triangle starts:
%e [0] 1;
%e [1] 0, 1;
%e [2] 0, 3, 1;
%e [3] 0, 4, 6, 1;
%e [4] 0, 7, 17, 9, 1;
%e [5] 0, 6, 38, 39, 12, 1;
%e [6] 0, 12, 70, 120, 70, 15, 1;
%e [7] 0, 8, 116, 300, 280, 110, 18, 1;
%e [8] 0, 15, 185, 645, 885, 545, 159, 21, 1;
%e [9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;
%p P := proc(n, x) option remember; if n = 0 then 1 else
%p x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
%p Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
%p seq(Trow(n), n=0..9);
%p # second Maple program:
%p T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _Alois P. Heinz_, Feb 01 2021
%p # Uses function PMatrix from A357368.
%p PMatrix(10, NumberTheory:-sigma); # _Peter Luschny_, Oct 19 2022
%t T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
%t If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
%t With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 11 2021, after _Alois P. Heinz_ *)
%Y Columns k=0..2 give: A000007, A000203, A000385.
%Y Row sums are A180305.
%Y T(2n,n) gives A340993.
%Y Cf. A008298, A078521, A319933.
%K nonn,tabl
%O 0,5
%A _Peter Luschny_, Oct 03 2018
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