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A241885
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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 f(n).
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8
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1, -1, 1, 1, -3, -19, 79, 275, -2339, -11813, 14217, 95265, -4634445, -193814931, 131301607, 1315505395, -3890947599, -136146236611, 46949081169401, 124889801445461, -10635113572583999, -158812278992229461, 56918172351554857, 8484151253958927197
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OFFSET
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0,5
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COMMENTS
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For g(n) see A242225(n).
The old definition was "Numerator 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 and J. Int. Seq. 17 (2014) # 14.6.7.
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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.
E.g.f: sqrt(x/(exp(x)-1)); take numerators. - Peter Luschny, May 08 2014
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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)=-1.
For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=79.
1/1, -1/4, 1/48, 1/64, -3/1280, -19/3072, 79/86016, 275/49152, -2339/2949120, -11813/1310720, 14217/11534336 = A241885 / A242225
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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 -> numer(g(bernoulli, n));
seq(a(n), n=0..23); # Peter Luschny, May 07 2014
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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[Factor[g[k]], {k, 0, 15}] // TableForm
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CROSSREFS
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Cf. A242225, A126156, A242233.
Cf. also A222411/A222412, A350194/A350154.
Sequence in context: A027175 A093734 A099421 * A061171 A293561 A240286
Adjacent sequences: A241882 A241883 A241884 * A241886 A241887 A241888
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KEYWORD
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sign,frac
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AUTHOR
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Jitender Singh, May 01 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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