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A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].

%I A182928 #14 Mar 07 2016 02:39:32
%S A182928 1,1,-1,1,2,1,-3,-6,1,24,1,-10,30,-120,1,720,1,-35,-630,-5040,1,560,
%T A182928 40320,1,-126,22680,-362880,1,3628800,1,-462,11550,-92400,-1247400,
%U A182928 -39916800,1,479001600,1,-1716,97297200,-6227020800
%N A182928 Triangular array read by rows: [T(n,k),k=1..tau(n)] = [-n!/(d*(-(n/d)!)^d), d|n].
%C A182928 The number of terms in the n-th row is the number of divisors of n. The n-th row is (apart from sign) a subsequence of the column labeled "M_1" for n-1 in Abramowitz and Stegun, Handbook, p. 831.
%C A182928 Let s(n) be the sum of row n. The number of partitions of an n-set with distinct block sizes can be computed recursively as A007837(0) = 1 and A007837(n) = - Sum_{1<=k<=n} binomial(n-1,k-1)*s(k)*A007837(n-k).
%C A182928 Let t(n) be the sum of the absolute values of row n. The sum of multinomial coefficients can be computed recursively as A005651(0) = 1 and A005651(n) = Sum_{1<=k<=n} binomial(n-1,k-1)*t(k)*A005651(n-k).
%e A182928 The array starts with
%e A182928 [1] 1,
%e A182928 [2] 1,  -1,
%e A182928 [3] 1,   2,
%e A182928 [4] 1,  -3,   -6,
%e A182928 [5] 1,  24,
%e A182928 [6] 1, -10,   30,  -120,
%e A182928 [7] 1, 720,
%e A182928 [8] 1, -35,  -630, -5040,
%e A182928 [9] 1, 560, 40320,
%p A182928 A182928_row := proc(n) local d;
%p A182928 seq(-n!/(d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:
%t A182928 row[n_] := Table[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[row[n], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Jul 29 2013 *)
%Y A182928 Cf. A076901, A132958, A132959, A132960, A132962.
%K A182928 sign,tabf
%O A182928 1,5
%A A182928 _Peter Luschny_, Apr 13 2011