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%I #17 Jul 12 2019 03:32:44
%S 1,0,1,2,0,-1,1,2,1,2,0,0,1,2,1,1,-2,0,1,3,1,5,6,0,0,-1,-1,2,1,5,5,-2,
%T 0,-2,-3,2,2,9,10,0,1,4,3,0,4,0,2,9,9,-3,1,3,-2,-7,2,0,3,14,16,0,2,6,
%U -1,-9,2,0,3,15,17,-2,1,8,19,10,-6,4,0,-1,0,15,22,0,1,9,21,7,-13,2,0,-2,-4,11,20,-4,2,15,33,14,-15,3,0,-4,-10,10,28,0,3
%N Triangle of generalized sum of divisors function, read by rows.
%C Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).
%H P. A. MacMahon, <a href="https://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 (1919), 75-113; Coll. Papers II, pp. 303-341.
%F 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)) = Sum_n T(n, k)*q^n.
%e Triangle turned on its side begins:
%e 1 0 2 -1 2 0 2 -2 3 0 2 ...
%e 1 0 1 2 1 1 1 6 -1 ...
%e 1 0 1 0 5 -1 5 ...
%t max = 27(*rows*); t[n_, k_] := Module[{m, mm, q, s}, mm = Array[m, k]; s = Sum[q^Total[mm]/Times @@ (1+q^mm), Evaluate[Sequence @@ Transpose[{mm, Join[{1}, Most[mm]+1], max-Range[k-1, 0, -1]}]]]; SeriesCoefficient[s, {q, 0, n}]]; Table[Print[an = Table[t[n, k], {k, Floor[(Sqrt[8*n+1]-1)/2], 1, -1}]]; an, {n, 1, max}] // Flatten (* _Jean-François Alcover_, Jan 21 2014 *)
%Y Diagonals give A048272, A060185, A060186.
%Y Cf. A060043, A060044, A060047, A060177.
%K sign,tabf,easy,nice
%O 1,4
%A _N. J. A. Sloane_, Mar 20 2001
%E More terms from _Vladeta Jovovic_, Sep 20 2007
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