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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). 6
1, 4, 48, 64, 1280, 3072, 86016, 49152, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 12079595520, 3221225472, 73014444032, 51539607552, 137095356088320, 5772436045824, 3809807790243840, 725677674332160, 2023101395107840, 3166593487994880 (list; graph; refs; listen; history; text; internal format)



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".


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.


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.


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.


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);


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


Cf. A241885.

Cf. also A222411/A222412, A350194/A350154.

Sequence in context: A358889 A225987 A178429 * A157818 A048608 A275033

Adjacent sequences: A242222 A242223 A242224 * A242226 A242227 A242228




Jitender Singh, May 08 2014


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



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Last modified February 5 17:56 EST 2023. Contains 360086 sequences. (Running on oeis4.)