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A060186 Generalized sum of divisors function: third diagonal of A060184.

%I A060186 #19 Jul 12 2019 02:52:08
%S A060186 1,0,1,0,5,-1,5,-2,9,3,9,-2,14,-1,15,10,15,7,11,14,10,26,20,28,2,41,
%T A060186 -5,63,-21,82,-5,91,-49,122,-46,139,-84,165,-74,240,-147,242,-142,290,
%U A060186 -217,333,-189,378,-284,463,-290,508,-408,560,-377
%N A060186 Generalized sum of divisors function: third diagonal of A060184.
%H A060186 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 A060186 G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum((x^n/(1+x^(n)))^i,n=1..inf), i=1..3. - _Vladeta Jovovic_, Sep 20 2007
%p A060186 mufact := proc(k,sumax) local res,c,i,j,isord,s ; res := [] ; for s from k*(k+1)/2 to sumax do c := combinat[composition](s,k) ; for j from 1 to nops(c) do isord := true ; for i from 2 to nops(op(j,c)) do if op(i,op(j,c))<= op(i-1,op(j,c)) then isord := false ; fi ; od ; if isord then res := [op(res),op(j,c)] ; fi ; od ; od ; RETURN(res) ; end: qm := proc(gfpart,n) local f,i ; f := q^add(op(i,gfpart),i=1..nops(gfpart)) ; for i from 1 to nops(gfpart) do f := taylor(f/(1+q^op(i,gfpart)),q=0,n+1) ; od ; RETURN(f) ; end: A060186 := proc(n) local k,ms,gf,gfpart,i ; k := 3 ; ms := mufact(k,n) ; gf := 0; for i from 1 to nops(ms) do gfpart := op(i,ms) ; gf := taylor(gf+qm(gfpart,n),q=0,n+1) ; od ; RETURN(gf) ; end: nmax := 60 : a := A060186(nmax) : for n from 6 to nmax do printf("%d, ",coeftayl(a,q=0,n)) ; od ; # _R. J. Mathar_, Jun 26 2007
%t A060186 max = 60; t[i_] := Sum[(x^n/(1 + x^(n)))^i, {n, 1, max}]; s = Series[(t[1]^3 - 3*t[1]*t[2] + 2*t[3])/6, {x, 0, max+1}]; a[n_] := SeriesCoefficient[s, {x, 0, n}]; Table[a[n], {n, 6, max}] (* _Jean-François Alcover_, Jan 17 2014, after _Vladeta Jovovic_ *)
%K A060186 easy,sign
%O A060186 3,5
%A A060186 _N. J. A. Sloane_, Mar 19 2001
%E A060186 More terms from _R. J. Mathar_, Jun 26 2007