cp's OEIS Frontend

A241885 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).

%I A241885 #58 Oct 10 2024 15:30:58
%S A241885 1,-1,1,1,-3,-19,79,275,-2339,-11813,14217,95265,-4634445,-193814931,
%T A241885 131301607,1315505395,-3890947599,-136146236611,46949081169401,
%U A241885 124889801445461,-10635113572583999,-158812278992229461,56918172351554857,8484151253958927197
%N A241885 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).
%C A241885 For g(n) see A242225(n).
%C A241885 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".
%C A241885 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 A241885 Robert Israel, <a href="/A241885/b241885.txt">Table of n, a(n) for n = 0..479</a>
%H A241885 David Broadhurst, <a href="/A241885/a241885.txt">Relations between A241885/A242225, A222411/A222412, and A350194/A350154.</a>
%H A241885 Jitender Singh, <a href="http://arxiv.org/abs/1402.0065">On an arithmetic convolution</a>, arXiv:1402.0065 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Singh/singh8.html">J. Int. Seq. 17 (2014) # 14.6.7</a>.
%F A241885 Theorem: a(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 A241885 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 A241885 g(0) = f(0)^(1/m);
%F A241885 g(1) = f(1)/(m*g(0)^(m-1));
%F A241885 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 A241885 This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.
%F A241885 E.g.f: sqrt(x/(exp(x)-1)); take numerators. - _Peter Luschny_, May 08 2014
%F A241885 a(n) = numerator(Sum_{k=0..n} binomial(-1/2,k)*binomial(n+1/2,n-k)*Stirling2(n+k,k)/binomial(n+k,k)). - _Tani Akinari_, Oct 08 2024
%e A241885 For n=1, B_1=-1/2 and B_1^(1/2)=-1/4 so a(1)=-1.
%e A241885 For n=6, B_6=1/6 and B_6^(1/2)=79/86016 so a(6)=79.
%e A241885 1/1, -1/4, 1/48, 1/64, -3/1280, -19/3072, 79/86016, 275/49152, -2339/2949120, -11813/1310720, 14217/11534336 = A241885 / A242225.
%p A241885 g := proc(f, n) option remember; local g0, m; g0 := sqrt(f(0));
%p A241885 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 A241885 a := n -> numer(g(bernoulli, n));
%p A241885 seq(a(n), n = 0..23); # _Peter Luschny_, May 07 2014
%t A241885 a := 1
%t A241885 g[0] := Sqrt[f[0]]
%t A241885 f[k_] := BernoulliB[k]
%t A241885 g[1] := f[1]/(2 g[0]^1);
%t A241885 g[k_] := (f[k] - Sum[Binomial[k, m] g[m] g[k - m], {m, 1, k - 1}])/(2 g[0])
%t A241885 Table[Factor[g[k]], {k, 0, 15}] // TableForm
%t A241885 (* Alternative: *)
%t A241885 Table[Numerator@NorlundB[n, 1/2, 0], {n, 0, 23}]  (* _Peter Luschny_, Feb 18 2024 *)
%o A241885 (PARI) a(n)=numerator(sum(k=0,n,binomial(-1/2,k)*binomial(n+1/2,n-k)*stirling(n+k,k,2)/binomial(n+k,k))) \\ _Tani Akinari_, Oct 08 2024
%Y A241885 Cf. A242225 (denominators), A126156, A242233.
%Y A241885 Cf. also A222411/A222412, A350194/A350154.
%Y A241885 Cf. A370416/A370417.
%K A241885 sign,frac
%O A241885 0,5
%A A241885 _Jitender Singh_, May 01 2014
%E A241885 Simpler definition from _N. J. A. Sloane_, Apr 24 2022 at the suggestion of _David Broadhurst_