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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.

%I A319083 #35 Feb 08 2025 00:35:35
%S A319083 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 A319083 8,116,300,280,110,18,1,0,15,185,645,885,545,159,21,1,0,13,258,1261,
%U A319083 2364,2095,942,217,24,1,0,18,384,2262,5586,6713,4281,1498,284,27,1
%N 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.
%C A319083 Column k is the k-fold self-convolution of sigma (A000203). - _Alois P. Heinz_, Feb 01 2021
%C A319083 For fixed k, Sum_{j=1..n} T(j,k) ~ Pi^(2*k) * n^(2*k) / (6^k * (2*k)!). - _Vaclav Kotesovec_, Sep 20 2024
%H A319083 Alois P. Heinz, <a href="/A319083/b319083.txt">Rows n = 0..200, flattened</a>
%F A319083 The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
%F A319083 p(n, x) = x*Sum_{k=0..n-1} sigma(n-k)*p(k, x).
%F A319083 Sum_{k=0..n} (-1)^k * T(n,k) = A283334(n). - _Alois P. Heinz_, Feb 07 2025
%e A319083 Triangle starts:
%e A319083 [0] 1;
%e A319083 [1] 0,  1;
%e A319083 [2] 0,  3,   1;
%e A319083 [3] 0,  4,   6,    1;
%e A319083 [4] 0,  7,  17,    9,    1;
%e A319083 [5] 0,  6,  38,   39,   12,    1;
%e A319083 [6] 0, 12,  70,  120,   70,   15,   1;
%e A319083 [7] 0,  8, 116,  300,  280,  110,  18,   1;
%e A319083 [8] 0, 15, 185,  645,  885,  545, 159,  21,  1;
%e A319083 [9] 0, 13, 258, 1261, 2364, 2095, 942, 217, 24, 1;
%p A319083 P := proc(n, x) option remember; if n = 0 then 1 else
%p A319083 x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
%p A319083 Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
%p A319083 seq(Trow(n), n=0..9);
%p A319083 # second Maple program:
%p A319083 T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A319083       `if`(k=1, `if`(n=0, 0, numtheory[sigma](n)), (q->
%p A319083        add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A319083     end:
%p A319083 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Feb 01 2021
%p A319083 # Uses function PMatrix from A357368.
%p A319083 PMatrix(10, NumberTheory:-sigma); # _Peter Luschny_, Oct 19 2022
%t A319083 T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
%t A319083      If[k == 1, If[n == 0, 0, DivisorSigma[1, n]],
%t A319083      With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
%t A319083 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 11 2021, after _Alois P. Heinz_ *)
%Y A319083 Columns k=0..6 give: A000007, A000203, A000385, A374951, A374977, A374978, A374979.
%Y A319083 Row sums are A180305.
%Y A319083 T(2n,n) gives A340993.
%Y A319083 Cf. A008298, A078521, A283334, A319933.
%K A319083 nonn,tabl
%O A319083 0,5
%A A319083 _Peter Luschny_, Oct 03 2018