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

1,2

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.

Links

Table of n, a(n) for n=1..39.

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.

Example

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.

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 *)

Prog

(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))}

Crossrefs

Cf. A053657.

Cf. A163972 (MC polynomials).

Sequence in context: A072424 A089250 A155681 * A221710 A011504 A357467

Adjacent sequences: A075261 A075262 A075263 * A075265 A075266 A075267

Keyword

frac,nonn,tabl

Author

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

Status

approved