A060186 Generalized sum of divisors function: third diagonal of A060184.
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, -5, 63, -21, 82, -5, 91, -49, 122, -46, 139, -84, 165, -74, 240, -147, 242, -142, 290, -217, 333, -189, 378, -284, 463, -290, 508, -408, 560, -377
Offset: 3
Links
- P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113; Coll. Papers II, pp. 303-341.
Programs
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Maple
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
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Mathematica
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 *)
Formula
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
Extensions
More terms from R. J. Mathar, Jun 26 2007