A078521 Signed triangle of D'Arcais numbers (A008298) : coefficients of r in the polynomials generated by the series coefficients of z^n in Product[(1-z^k)^r, {k,1,Inf}]*(n!).

1, 0, -1, 0, -3, 1, 0, -8, 9, -1, 0, -42, 59, -18, 1, 0, -144, 450, -215, 30, -1, 0, -1440, 3394, -2475, 565, -45, 1, 0, -5760, 30912, -28294, 9345, -1225, 63, -1, 0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1, 0, -524160, 3032208, -4335596, 2341332, -579369, 69552, -4074, 108, -1, 0, -6531840

Offset: 1

Author

Wouter Meeussen, Jan 07 2003

Keywords

• easy
• sign
• tabl

Comments

Also the Bell transform of -A038048(n+1). For the definition of the Bell transform see A264428. - Peter Luschny, Jan 26 2016

Examples

			The z-expansion of Product[(1-z^k)^r, {k,1,3}] is 1 - r*z + ((-3+r)*r*z^2)/2 -(r*(8-9*r +r^2)*z^3)/6, so the third row of the triangle is 0,-8,9,-1.
Triangle begins
1,
0, -1,
0, -3, 1,
0, -8, 9, -1,
0, -42, 59, -18, 1,
0, -144, 450, -215, 30, -1,
0, -1440, 3394, -2475, 565, -45, 1,
0, -5760, 30912, -28294, 9345, -1225, 63, -1,
0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1
...
		

Crossrefs

Cf. A008298, A010815, A038048.

Programs

Maple

# The function BellMatrix is defined in A264428.
BellMatrix(n -> -n!*numtheory:-sigma(n+1), 9); # Peter Luschny, Jan 26 2016
# Alternative:
P := proc(n, x) option remember; if n = 0 then 1 else
-(1/n)*x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:
Trow := n -> seq(n!*coeff(P(n, x), x, k), k=0..n):
seq(Trow(n), n=0..9); # Peter Luschny, Oct 03 2018

Mathematica

w=16;(CoefficientList[ #, r]&/@ CoefficientList[Series[Product[(1-z^k)^r, {k, 1, w}], {z, 0, w}], z])Range[0, w]!
(* Second program: *)
BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
B = BellMatrix[Function[n, -n!*DivisorSigma[1, n + 1]], rows = 12];
Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 28 2018, after Peter Luschny *)

Formula

See Mathematica line.

Row sums give A010815 * n!.