login
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!).
2

%I #19 Oct 03 2018 08:27:14

%S 1,0,-1,0,-3,1,0,-8,9,-1,0,-42,59,-18,1,0,-144,450,-215,30,-1,0,-1440,

%T 3394,-2475,565,-45,1,0,-5760,30912,-28294,9345,-1225,63,-1,0,-75600,

%U 293292,-340116,147889,-27720,2338,-84,1,0,-524160,3032208,-4335596,2341332,-579369,69552,-4074,108,-1,0,-6531840

%N 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!).

%C Also the Bell transform of -A038048(n+1). For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 26 2016

%F See Mathematica line.

%F Row sums give A010815 * n!.

%e 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.

%e Triangle begins

%e 1,

%e 0, -1,

%e 0, -3, 1,

%e 0, -8, 9, -1,

%e 0, -42, 59, -18, 1,

%e 0, -144, 450, -215, 30, -1,

%e 0, -1440, 3394, -2475, 565, -45, 1,

%e 0, -5760, 30912, -28294, 9345, -1225, 63, -1,

%e 0, -75600, 293292, -340116, 147889, -27720, 2338, -84, 1

%e ...

%p # The function BellMatrix is defined in A264428.

%p BellMatrix(n -> -n!*numtheory:-sigma(n+1), 9); # _Peter Luschny_, Jan 26 2016

%p # Alternative:

%p P := proc(n, x) option remember; if n = 0 then 1 else

%p -(1/n)*x*add(numtheory:-sigma(n-k)*P(k,x), k=0..n-1) fi end:

%p Trow := n -> seq(n!*coeff(P(n, x), x, k), k=0..n):

%p seq(Trow(n), n=0..9); # _Peter Luschny_, Oct 03 2018

%t w=16;(CoefficientList[ #, r]&/@ CoefficientList[Series[Product[(1-z^k)^r, {k, 1, w}], {z, 0, w}], z])Range[0, w]!

%t (* Second program: *)

%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];

%t B = BellMatrix[Function[n, -n!*DivisorSigma[1, n + 1]], rows = 12];

%t Table[B[[n, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)

%Y Cf. A008298, A010815, A038048.

%K easy,sign,tabl

%O 1,5

%A _Wouter Meeussen_, Jan 07 2003