A242225 Write the coefficient of x^n/n! in the expansion of (x/(exp(x)-1))^(1/2) as f(n)/g(n); sequence gives g(n).

1, 4, 48, 64, 1280, 3072, 86016, 49152, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 12079595520, 3221225472, 73014444032, 51539607552, 137095356088320, 5772436045824, 3809807790243840, 725677674332160, 2023101395107840, 3166593487994880

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

Table of n, a(n) for n=0..23.

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

Cf. A241885.

Cf. also A222411/A222412, A350194/A350154.

Cf. A370416/A370417.

Sequence in context: A225987 A178429 A370417 * A157818 A362402 A048608

Adjacent sequences: A242222 A242223 A242224 * A242226 A242227 A242228

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