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

1, 1, -1, 1, 2, 1, -3, -6, 1, 24, 1, -10, 30, -120, 1, 720, 1, -35, -630, -5040, 1, 560, 40320, 1, -126, 22680, -362880, 1, 3628800, 1, -462, 11550, -92400, -1247400, -39916800, 1, 479001600, 1, -1716, 97297200, -6227020800

Offset: 1

Author

Peter Luschny, Apr 13 2011

Keywords

• sign
• tabf

Comments

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.

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).

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).

Examples

			The array starts with
[1] 1,
[2] 1,  -1,
[3] 1,   2,
[4] 1,  -3,   -6,
[5] 1,  24,
[6] 1, -10,   30,  -120,
[7] 1, 720,
[8] 1, -35,  -630, -5040,
[9] 1, 560, 40320,
		

Crossrefs

Cf. A076901, A132958, A132959, A132960, A132962.

Programs

Maple

A182928_row := proc(n) local d;
seq(-n!/(d*(-(n/d)!)^d), d = numtheory[divisors](n)) end:

Mathematica

row[n_] := Table[ -n!/(d*(-(n/d)!)^d), {d, Divisors[n]}]; Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)