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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). 7
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A237656

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

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Last modified September 21 22:07 EDT 2019. Contains 327283 sequences. (Running on oeis4.)