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A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.

%I A320019 #35 Jul 27 2024 09:40:27
%S A320019 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,
%T A320019 78,92,50,12,1,0,4,37,132,216,175,72,14,1,0,3,44,209,440,490,298,98,
%U A320019 16,1,0,4,58,306,814,1172,972,469,128,18,1
%N A320019 Coefficients of polynomials related to the number of divisors, triangle read by rows, T(n,k) for 0 <= k <= n.
%C A320019 Column k is the k-fold self-convolution of tau (A000005). - _Alois P. Heinz_, Feb 01 2021
%H A320019 Alois P. Heinz, <a href="/A320019/b320019.txt">Rows n = 0..200, flattened</a>
%F A320019 The polynomials are defined by recurrence: p(0,x) = 1 and for n > 0 by
%F A320019 p(n, x) = x*Sum_{k=0..n-1} tau(n-k)*p(k, x).
%F A320019 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.
%F A320019 T(n,k) = [x^n] (Sum_{j>=1} tau(j)*x^j)^k. - _Alois P. Heinz_, Feb 14 2021
%e A320019 Triangle starts:
%e A320019 [0] 1
%e A320019 [1] 0, 1
%e A320019 [2] 0, 2,  1
%e A320019 [3] 0, 2,  4,   1
%e A320019 [4] 0, 3,  8,   6,   1
%e A320019 [5] 0, 2, 14,  18,   8,   1
%e A320019 [6] 0, 4, 20,  41,  32,  10,   1
%e A320019 [7] 0, 2, 28,  78,  92,  50,  12,  1
%e A320019 [8] 0, 4, 37, 132, 216, 175,  72, 14,  1
%e A320019 [9] 0, 3, 44, 209, 440, 490, 298, 98, 16, 1
%p A320019 P := proc(n, x) option remember; if n = 0 then 1 else
%p A320019 x*add(numtheory:-tau(n-k)*P(k,x), k=0..n-1) fi end:
%p A320019 Trow := n -> seq(coeff(P(n, x), x, k), k=0..n):
%p A320019 seq(lprint([n], Trow(n)), n=0..9);
%p A320019 # second Maple program:
%p A320019 T:= proc(n, k) option remember; `if`(k=0, `if`(n=0, 1, 0),
%p A320019       `if`(k=1, `if`(n=0, 0, numtheory[tau](n)), (q->
%p A320019        add(T(j, q)*T(n-j, k-q), j=0..n))(iquo(k, 2))))
%p A320019     end:
%p A320019 seq(seq(T(n, k), k=0..n), n=0..12);  # _Alois P. Heinz_, Feb 01 2021
%p A320019 # Uses function PMatrix from A357368.
%p A320019 PMatrix(10, NumberTheory:-tau); # _Peter Luschny_, Oct 19 2022
%t A320019 T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],
%t A320019      If[k == 1, If[n == 0, 0, DivisorSigma[0, n]],
%t A320019      With[{q = Quotient[k, 2]}, Sum[T[j, q]*T[n-j, k-q], {j, 0, n}]]]];
%t A320019 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 11 2021, after _Alois P. Heinz_ *)
%Y A320019 Columns k=0-4 give: A000007, A000005, A055507, A191829, A375002.
%Y A320019 Row sums are A129921.
%Y A320019 T(2n,n) gives A340992.
%Y A320019 Cf. A319083.
%K A320019 nonn,tabl
%O A320019 0,5
%A A320019 _Peter Luschny_, Oct 03 2018