A075264 Triangle of numerators of coefficients, where the n-th row forms the polynomial in z, P(n,z), that is the coefficient of x^n in {-log(1-x)/x}^z, for n > 0. The denominator for all the terms in the n-th row is A053657(n).

1, 5, 3, 6, 5, 1, 502, 485, 150, 15, 760, 802, 305, 50, 3, 152696, 171150, 73801, 15435, 1575, 63, 252336, 295748, 139020, 33817, 4515, 315, 9, 51360816, 62333204, 31231500, 8437975, 1334760, 124110, 6300, 135, 88864128, 110941776, 58415444

Offset: 1

Author

Paul D. Hanna, Sep 15 2002; revised Jun 27 2005

Keywords

• frac
• nonn
• tabl

Comments

Each n-th row polynomial, P(n,z), has a trivial zero at z = 0; for odd rows, P(2n+1,z) also has zeros at z = -2n, z = -(2n+1), for n > 0.

Examples

			P(1,z) = z/2,
P(2,z) = (5z + 3z^2)/24,
P(3,z) = (6z + 5z^2 + z^3)/48,
P(4,z) = (502z + 485z^2 + 150z^3 + 15z^4)/5760,
P(5,z) = (760z + 802z^2 + 305z^3 + 50z^4 +3z^5)/11520,
P(6,z) = (152696z + 171150z^2 + 73801z^3 + 15435z^4 + 1575z^5
+ 63z^6)/2903040,
P(7,z) = (252336z + 295748z^2 + 139020z^3 + 33817z^4 + 4515z^5
+ 315z^6 + 9z^7)/5806080,
P(8,z) = (51360816z + 62333204z^2 + 31231500z^3 + 8437975z^4
+ 1334760z^5 + 124110z^6 + 6300z^7 + 135z^8)/1393459200.
		

Crossrefs

Cf. A053657.

Cf. A163972 (MC polynomials).

Programs

Maple

nmax:=8; A053657 := proc(n) local P, p, q, s, r; P := select(isprime, [$2..n]); r:=1; for p in P do s := 0; q := p-1; do if q > (n-1) then break fi; s := s + iquo(n-1, q); q := q*p; od; r := r * p^s; od; r end: f(z) := convert(series((-ln(1-x)/x)^z, x, nmax+2), polynom): for n from 1 to nmax do f(n) := A053657(n+1)*coeff(f(z), x, n) od: for n from 1 to nmax do for m from 1 to n do a(n, m) := coeff(f(n), z, m) od: od: seq(seq(a(n, m), m=1..n), n=1..nmax);  # Johannes W. Meijer, Jun 08 2009, revised Nov 25 2012

Mathematica

rows = 9; A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; (Rest[CoefficientList[#, z]]& /@ Rest @ CoefficientList[(-Log[1-x]/x)^z + O[x]^(rows+1), x]) * Array[A053657, rows, 2] // Flatten (* Jean-François Alcover, Nov 22 2016 *)

PARI

{T(n,k)=local(X=x+x^2*O(x^n)); D=1;for(j=0,n,D=lcm(D,denominator( polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^j),j,z),n,x)))); return(D*polcoeff(polcoeff((-log(1-X)/x)^z+z*O(z^k),k,z),n,x))}

Formula

The n-th row polynomials, P(n, z), satisfy 1 + Sum_{n>=1} P(n, z) x^n = (Sum_{k>=1} x^(k-1)/k)^z.