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A060043 Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.

%I A060043 #33 Jul 21 2025 09:05:32
%S A060043 1,3,1,4,3,7,9,6,1,15,12,3,30,8,9,45,15,22,67,13,1,42,99,18,3,81,135,
%T A060043 12,9,140,175,28,22,231,231,14,51,351,306,24,1,97,551,354,24,3,188,
%U A060043 783,465,31,9,330,1134,540,18,22,568,1546,681,39,51,918,2142,765,20
%N A060043 Triangle T(n,k), n >= 1, k >= 1, of generalized sum of divisors function, read by rows.
%C A060043 Lengths of rows are 1 1 2 2 2 3 3 3 3 ... (A003056).
%H A060043 Alois P. Heinz, <a href="/A060043/b060043.txt">Rows n = 1..1000, flattened</a>
%H A060043 G. E. Andrews and S. C. F. Rose, <a href="http://arxiv.org/abs/1010.5769">MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms</a>, arXiv:1010.5769 [math.NT], 2010.
%H A060043 P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
%F A060043 T(n, 1) = sum of divisors of n (A000203), T(n, k) = sum of s_1*s_2*...*s_k where s_1, s_2, ..., s_k are such that s_1*m_1 + s_2*m_2 + ... + s_k*m_k = n and the sum is over all such k-partitions of n.
%F A060043 G.f. for k-th diagonal (the k-th row of the sideways triangle shown in the example): Sum_{ m_1 < m_2 < ... < m_k} q^(m_1+m_2+...+m_k)/((1-q^m_1)*(1-q^m_2)*...*(1-q^m_k))^2 = Sum_n T(n, k)*q^n.
%F A060043 G.f. for k-th diagonal: (-1)^k * (1/(2*k+1)) * ( Sum_{j>=k} (-1)^j * (2*j+1) * binomial(j+k,2*k) * q^(j*(j+1)/2) ) / ( Sum_{j>=0} (-1)^j * (2*j+1) * q^(j*(j+1)/2) ). - _Seiichi Manyama_, Sep 15 2023
%e A060043 Triangle turned on its side begins:
%e A060043 1 3 4 7 6 12  8 15 13 18 ...
%e A060043     1 3 9 15 30 45 67 99 ...
%e A060043            1  3  9 22 42 ...
%e A060043                        1 ...
%e A060043 For example, T(6,2) = 15.
%p A060043 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A060043       b(n, i-1)+add(expand(b(n-i*j, i-1)*j*x), j=1..n/i)))
%p A060043     end:
%p A060043 T:= n-> (p-> seq(coeff(p, x, degree(p)-i), i=0..degree(p)-1))(b(n$2)):
%p A060043 seq(T(n), n=1..20);  # _Alois P. Heinz_, Jul 21 2025
%t A060043 Clear[diag, m]; nmax = 19; kmax = Floor[(Sqrt[8*nmax+1]-1)/2]; m[0] = 0; diag[k_] := diag[k] = Sum[q^(Sum[m[i], {i, 1, k}])/(Times @@ (1 - q^Array[m, k]))^2, Sequence @@ Table[{m[j], m[j-1]+1, nmax}, {j, 1, k}] // Evaluate] + O[q]^(nmax+1) // CoefficientList[#, q]&; Table[ Select[ Table[diag[k][[j+1]], {k, 1, kmax}], IntegerQ[#] && # > 0&] // Reverse, {j, 1, nmax}] // Flatten (* _Jean-François Alcover_, Jul 18 2017 *)
%Y A060043 Diagonals give A000203, A002127, A002128, A365664, A365665.
%Y A060043 Cf. A010816, A060044, A060047.
%K A060043 nonn,tabf,easy,look,nice
%O A060043 1,2
%A A060043 _N. J. A. Sloane_, Mar 19 2001
%E A060043 More terms from _Naohiro Nomoto_, Jan 24 2002