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A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.

%I A124433 #22 Aug 25 2020 09:45:55
%S A124433 1,0,-1,0,-1,0,-1,1,0,-1,0,-1,2,0,-1,0,-1,2,-1,0,-1,1,0,-1,2,0,-1,0,
%T A124433 -1,4,-3,0,-1,0,-1,2,0,-1,2,0,-1,3,-3,1,0,-1,0,-1,4,-3,0,-1,0,-1,4,-3,
%U A124433 0,-1,2,0,-1,2,0,-1,0,-1,6,-9,4,0,-1,1,0,-1,2,0,-1,2,-1,0,-1,4,-3,0,-1,0,-1,6,-6,0,-1,0,-1,4,-6,4,-1,0,-1,2,0,-1,2,0,-1
%N A124433 Irregular array {a(n,m)} read by rows where (sum{n>=1} sum{m=1 to A001222(n)+1} a(n,m)*y^m/n^x) = 1/(zeta(x)-1+1/y) for all x and y where the double sum converges.
%C A124433 Row n has A001222(n)+1 terms. The polynomial P_n(y) = (sum{m=1 to A001222(n)+1} a(n,m)*y^m) is a generalization of the Mobius (Moebius) function, where P_n(1) = A008683(n).
%C A124433 From _Gus Wiseman_, Aug 24 2020: (Start)
%C A124433 Up to sign, also the number of strict length-k chains of divisors from n to 1, 1 <= k <= 1 + A001222(n). For example, row n = 36 counts the following chains (empty column indicated by dot):
%C A124433   .  36/1  36/2/1   36/4/2/1   36/12/4/2/1
%C A124433            36/3/1   36/6/2/1   36/12/6/2/1
%C A124433            36/4/1   36/6/3/1   36/12/6/3/1
%C A124433            36/6/1   36/9/3/1   36/18/6/2/1
%C A124433            36/9/1   36/12/2/1  36/18/6/3/1
%C A124433            36/12/1  36/12/3/1  36/18/9/3/1
%C A124433            36/18/1  36/12/4/1
%C A124433                     36/12/6/1
%C A124433                     36/18/2/1
%C A124433                     36/18/3/1
%C A124433                     36/18/6/1
%C A124433                     36/18/9/1
%C A124433 (End)
%H A124433 Mohammad K. Azarian, <a href="https://www.jstor.org/stable/2686918">A Double Sum, Problem 440</a>, College Mathematics Journal, Vol. 21, No. 5, Nov. 1990, p. 424. <a href="https://www.jstor.org/stable/2686609">Solution</a> published in Vol. 22. No. 5, Nov. 1991, pp. 448-449.
%F A124433 a(1,1)=1. a(n,1) = 0 for n>=2. a(n,m+1) = -sum{k|n,k < n} a(k,m), where, for the purpose of this sum, a(k,m) = 0 if m > A001222(k)+1.
%e A124433 1/(zeta(x) - 1 + 1/y) = y - y^2/2^x - y^2/3^x + ( - y^2 + y^3)/4^x - y^2/5^x + ( - y^2 + 2y^3)/6^x - y^2/7^x + ...
%e A124433 From _Gus Wiseman_, Aug 24 2020: (Start)
%e A124433 The sequence of rows begins:
%e A124433      1: 1              16: 0 -1 3 -3 1     31: 0 -1
%e A124433      2: 0 -1           17: 0 -1            32: 0 -1 4 -6 4 -1
%e A124433      3: 0 -1           18: 0 -1 4 -3       33: 0 -1 2
%e A124433      4: 0 -1 1         19: 0 -1            34: 0 -1 2
%e A124433      5: 0 -1           20: 0 -1 4 -3       35: 0 -1 2
%e A124433      6: 0 -1 2         21: 0 -1 2          36: 0 -1 7 -12 6
%e A124433      7: 0 -1           22: 0 -1 2          37: 0 -1
%e A124433      8: 0 -1 2 -1      23: 0 -1            38: 0 -1 2
%e A124433      9: 0 -1 1         24: 0 -1 6 -9 4     39: 0 -1 2
%e A124433     10: 0 -1 2         25: 0 -1 1          40: 0 -1 6 -9 4
%e A124433     11: 0 -1           26: 0 -1 2          41: 0 -1
%e A124433     12: 0 -1 4 -3      27: 0 -1 2 -1       42: 0 -1 6 -6
%e A124433     13: 0 -1           28: 0 -1 4 -3       43: 0 -1
%e A124433     14: 0 -1 2         29: 0 -1            44: 0 -1 4 -3
%e A124433     15: 0 -1 2         30: 0 -1 6 -6       45: 0 -1 4 -3
%e A124433 (End)
%t A124433 f[l_List] := Block[{n = Length[l] + 1, c},c = Plus @@ Last /@ FactorInteger[n];Append[l, Prepend[ -Plus @@ Pick[PadRight[ #, c] & /@ l, Mod[n, Range[n - 1]], 0],0]]];Nest[f, {{1}}, 34] // Flatten(* _Ray Chandler_, Feb 13 2007 *)
%t A124433 chnsc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]];
%t A124433 Table[(-1)^k*Length[Select[chnsc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* _Gus Wiseman_, Aug 24 2020 *)
%Y A124433 A008480 gives rows ends (up to sign).
%Y A124433 A008683 gives row sums (the Moebius function).
%Y A124433 A073093 gives row lengths.
%Y A124433 A074206 gives unsigned row sums.
%Y A124433 A097805 is the restriction to powers of 2 (up to sign).
%Y A124433 A251683 is the unsigned version with zeros removed.
%Y A124433 A334996 is the unsigned version (except with a(1) = 0).
%Y A124433 A334997 is an unsigned non-strict version.
%Y A124433 A337107 is the restriction to factorial numbers.
%Y A124433 A001055 counts factorizations.
%Y A124433 A001222 counts prime factors with multiplicity.
%Y A124433 A067824 counts strict chains of divisors starting with n.
%Y A124433 A074206 counts strict chains of divisors from n to 1.
%Y A124433 A122651 counts strict chains of divisors summing to n.
%Y A124433 A167865 counts strict chains of divisors > 1 summing to n.
%Y A124433 A253249 counts strict chains of divisors.
%Y A124433 A337105 counts strict chains of divisors from n! to 1.
%Y A124433 Cf. A000005, A001221, A002033, A124010, A167865.
%K A124433 sign,tabf
%O A124433 1,13
%A A124433 _Leroy Quet_, Dec 15 2006
%E A124433 Extended by _Ray Chandler_, Feb 13 2007