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A242225
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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).
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6
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1, 4, 48, 64, 1280, 3072, 86016, 49152, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 12079595520, 3221225472, 73014444032, 51539607552, 137095356088320, 5772436045824, 3809807790243840, 725677674332160, 2023101395107840, 3166593487994880
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OFFSET
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0,2
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COMMENTS
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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".
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LINKS
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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.
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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);
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MATHEMATICA
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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
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CROSSREFS
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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
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KEYWORD
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nonn,frac
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AUTHOR
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Jitender Singh, May 08 2014
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EXTENSIONS
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Simpler definition from N. J. A. Sloane, Apr 24 2022 at the suggestion of David Broadhurst.
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STATUS
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approved
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