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%I #27 Feb 18 2024 11:54:44
%S 1,4,48,64,1280,3072,86016,49152,2949120,1310720,11534336,4194304,
%T 1526726656,2348810240,12079595520,3221225472,73014444032,51539607552,
%U 137095356088320,5772436045824,3809807790243840,725677674332160,2023101395107840,3166593487994880
%N 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).
%C For f(n) see A241885(n).
%C 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".
%C 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
%H David Broadhurst, <a href="/A241885/a241885.txt">Relations between A241885/A242225, A222411/A222412, and A350194/A350154.</a>
%H Jitender Singh, <a href="http://arxiv.org/abs/1402.0065">On an arithmetic convolution</a>, arXiv:1402.0065 [math.NT], 2014.
%F 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).
%F 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:
%F g(0)= f(0)^{1/m};
%F g(1)= f(1)/(mg(0)^(m-1));
%F 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.
%F This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.
%e For n=1, B_1=-1/2 and B_1^(1/2)=-1/4 so a(1)=4.
%e For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=86016.
%p g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
%p 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:
%p a := n -> denom(g(bernoulli, n));
%p seq(a(n), n=0..23);
%t a := 1
%t g[0] := Sqrt[f[0]]
%t f[k_] := BernoulliB[k]
%t g[1] := f[1]/(2 g[0]^1);
%t g[k_] := (f[k] - Sum[Binomial[k, m] g[m] g[k - m], {m, 1, k - 1}])/(2 g[0])
%t Table[Denominator[Factor[g[k]]], {k, 0, 15}] // TableForm
%t (* Alternative: *)
%t Table[Denominator@NorlundB[n, 1/2, 0], {n, 0, 23}] (* _Peter Luschny_, Feb 18 2024 *)
%Y Cf. A241885.
%Y Cf. also A222411/A222412, A350194/A350154.
%Y Cf. A370416/A370417.
%K nonn,frac
%O 0,2
%A _Jitender Singh_, May 08 2014
%E Simpler definition from _N. J. A. Sloane_, Apr 24 2022 at the suggestion of _David Broadhurst_.
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