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

Author

Jitender Singh, May 08 2014

Keywords

• nonn
• frac

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

Examples

			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.
		

Links

• David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
• Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.

Crossrefs

Cf. A241885.

Cf. also A222411/A222412, A350194/A350154.

Cf. A370416/A370417.

Programs

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 *)

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=2.

This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.

Extensions

Simpler definition from N. J. A. Sloane, Apr 24 2022 at the suggestion of David Broadhurst.