OFFSET
0,2
COMMENTS
For f(n) see A241885(n).
The old definition was "Denominator of (B_n)^(1/2) in the Cauchy type product (sometimes known as binomial transform) where B_n is the n-th Bernoulli number".
The Nørlund polynomials N(a, n, x) with parameter a = 1/2 evaluated at x = 0 give the rational values. - Peter Luschny, Feb 18 2024
LINKS
David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.
FORMULA
Theorem: A241885(n)/A242225(n) = n!*A222411(n)/(A222412(n)*(-1)^n/(1-2*n)) = n!*A350194(n)/(A350154(n)*(2*n+1)). - David Broadhurst, Apr 23 2022 (see Link).
For any arithmetic function f and a positive integer k>1, define the k-th root of f to be the arithmetic function g such that g*g*...*g(k times)=f and is determined by the following recursive formula:
g(0)= f(0)^{1/m};
g(1)= f(1)/(mg(0)^(m-1));
g(k)= 1/(m g(0)^{m-1})*(f(k)-sum_{k_1+...+k_m=k,k_i<k} k!/( k_1!...k_m!)g(k_1)... g(k_m)), for k>=2.
This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.
EXAMPLE
For n=1, B_1=-1/2 and B_1^(1/2)=-1/4 so a(1)=4.
For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=86016.
MAPLE
g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
if n=0 then g0 else if n=1 then 0 else add(binomial(n, m)*g(f, m)*g(f, n-m), m=1..n-1) fi; (f(n)-%)/(2*g0) fi end:
a := n -> denom(g(bernoulli, n));
seq(a(n), n=0..23);
MATHEMATICA
a := 1
g[0] := Sqrt[f[0]]
f[k_] := BernoulliB[k]
g[1] := f[1]/(2 g[0]^1);
g[k_] := (f[k] - Sum[Binomial[k, m] g[m] g[k - m], {m, 1, k - 1}])/(2 g[0])
Table[Denominator[Factor[g[k]]], {k, 0, 15}] // TableForm
(* Alternative: *)
Table[Denominator@NorlundB[n, 1/2, 0], {n, 0, 23}] (* Peter Luschny, Feb 18 2024 *)
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Jitender Singh, May 08 2014
EXTENSIONS
Simpler definition from N. J. A. Sloane, Apr 24 2022 at the suggestion of David Broadhurst.
STATUS
approved