%I #22 Aug 25 2020 09:45:55
%S 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 -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 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 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 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 From _Gus Wiseman_, Aug 24 2020: (Start)
%C 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 . 36/1 36/2/1 36/4/2/1 36/12/4/2/1
%C 36/3/1 36/6/2/1 36/12/6/2/1
%C 36/4/1 36/6/3/1 36/12/6/3/1
%C 36/6/1 36/9/3/1 36/18/6/2/1
%C 36/9/1 36/12/2/1 36/18/6/3/1
%C 36/12/1 36/12/3/1 36/18/9/3/1
%C 36/18/1 36/12/4/1
%C 36/12/6/1
%C 36/18/2/1
%C 36/18/3/1
%C 36/18/6/1
%C 36/18/9/1
%C (End)
%H 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 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 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 From _Gus Wiseman_, Aug 24 2020: (Start)
%e The sequence of rows begins:
%e 1: 1 16: 0 -1 3 -3 1 31: 0 -1
%e 2: 0 -1 17: 0 -1 32: 0 -1 4 -6 4 -1
%e 3: 0 -1 18: 0 -1 4 -3 33: 0 -1 2
%e 4: 0 -1 1 19: 0 -1 34: 0 -1 2
%e 5: 0 -1 20: 0 -1 4 -3 35: 0 -1 2
%e 6: 0 -1 2 21: 0 -1 2 36: 0 -1 7 -12 6
%e 7: 0 -1 22: 0 -1 2 37: 0 -1
%e 8: 0 -1 2 -1 23: 0 -1 38: 0 -1 2
%e 9: 0 -1 1 24: 0 -1 6 -9 4 39: 0 -1 2
%e 10: 0 -1 2 25: 0 -1 1 40: 0 -1 6 -9 4
%e 11: 0 -1 26: 0 -1 2 41: 0 -1
%e 12: 0 -1 4 -3 27: 0 -1 2 -1 42: 0 -1 6 -6
%e 13: 0 -1 28: 0 -1 4 -3 43: 0 -1
%e 14: 0 -1 2 29: 0 -1 44: 0 -1 4 -3
%e 15: 0 -1 2 30: 0 -1 6 -6 45: 0 -1 4 -3
%e (End)
%t 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 chnsc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]];
%t Table[(-1)^k*Length[Select[chnsc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* _Gus Wiseman_, Aug 24 2020 *)
%Y A008480 gives rows ends (up to sign).
%Y A008683 gives row sums (the Moebius function).
%Y A073093 gives row lengths.
%Y A074206 gives unsigned row sums.
%Y A097805 is the restriction to powers of 2 (up to sign).
%Y A251683 is the unsigned version with zeros removed.
%Y A334996 is the unsigned version (except with a(1) = 0).
%Y A334997 is an unsigned non-strict version.
%Y A337107 is the restriction to factorial numbers.
%Y A001055 counts factorizations.
%Y A001222 counts prime factors with multiplicity.
%Y A067824 counts strict chains of divisors starting with n.
%Y A074206 counts strict chains of divisors from n to 1.
%Y A122651 counts strict chains of divisors summing to n.
%Y A167865 counts strict chains of divisors > 1 summing to n.
%Y A253249 counts strict chains of divisors.
%Y A337105 counts strict chains of divisors from n! to 1.
%Y Cf. A000005, A001221, A002033, A124010, A167865.
%K sign,tabf
%O 1,13
%A _Leroy Quet_, Dec 15 2006
%E Extended by _Ray Chandler_, Feb 13 2007