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%I #12 Nov 26 2019 04:18:25
%S 0,0,0,1,2,8,18,0,34,170,1643,3603,0,25118,139063,474559,284490,
%T 984006,6536387,24265729,18678366,96214018,277799290,1282283434,
%U 2077807072,1899874612,19252363859,44221482383,1967547352,29743945396,1265868622
%N Sum of proper divisors minus the number of proper divisors of Motzkin number A001006(n).
%C Note that, if a(n) != 0 then Motzkin number A001006(n) is a composite number (A002808), otherwise A001006(n) is a noncomposite number (A008578). See A152770.
%H Amiram Eldar, <a href="/A154349/b154349.txt">Table of n, a(n) for n = 0..201</a>
%F a(n) = A001065(A001006(n)) - A032741(A001006(n)) = A152770(A001006(n)).
%p with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(sigma(M(n))-M(n)-tau(M(n))+1, n = 0 .. 30); # _Emeric Deutsch_, Jan 12 2009
%t mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; diff[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Table[diff[mot[n]], {n, 0, 30}] (* _Amiram Eldar_, Nov 26 2019 *)
%Y Cf. A001006, A001065, A002808, A032741, A008578, A152770, A152981, A152982, A152983, A152988, A152990.
%K nonn
%O 0,5
%A _Omar E. Pol_, Jan 07 2009
%E Extended by _Emeric Deutsch_, Jan 12 2009