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).

1, -1, 1, 1, -3, -19, 79, 275, -2339, -11813, 14217, 95265, -4634445, -193814931, 131301607, 1315505395, -3890947599, -136146236611, 46949081169401, 124889801445461, -10635113572583999, -158812278992229461, 56918172351554857, 8484151253958927197

Offset: 0

Author

Jitender Singh, May 01 2014

Keywords

• sign
• frac

Comments

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".

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..479
• 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.

Crossrefs

Cf. A242225 (denominators), A126156, A242233.

Cf. also A222411/A222412, A350194/A350154.

Cf. A370416/A370417.

Programs

Maple

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

Mathematica

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
(* Alternative: *)
Table[Numerator@NorlundB[n, 1/2, 0], {n, 0, 23}]  (* Peter Luschny, Feb 18 2024 *)

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

Formula

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).

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)/(m*g(0)^(m-1));

g(k) = 1/(m*g(0)^(m-1))*(f(k) - Sum_{k_1+...+k_m=k,k_i= 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

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

Extensions

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