A222411 -id:A222411 - 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).

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

Jitender Singh, May 01 2014

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

			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.

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.

Cf. A242225 (denominators), A126156, A242233.
Cf. also A222411/A222412, A350194/A350154.
Cf. A370416/A370417.

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

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 *)

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

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

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

A222412 Denominators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

Original entry on oeis.org

1, 4, 32, 384, 10240, 40960, 61931520, 49545216, 7927234560, 475634073600, 1993133260800, 177167400960, 48753634065776640, 195014536263106560, 39002907252621312000, 842462796656620339200, 2204424056667635712000, 79359266040034885632000

Offset: 0

N. J. A. Sloane, Feb 28 2013

			The first few fractions are 1, -1/4, -1/32, 5/384, 7/10240, -19/40960, -869/61931520, 715/49545216, ... = A222411/A222412. - _Petros Hadjicostas_, May 14 2020

Alois P. Heinz, Table of n, a(n) for n = 0..300
David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011.
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130.
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.

Cf. A222411.

Cf. also A241885/A242225, A350194/A350154.

gf:= (x/(exp(x)-1))^(3/2)*exp(x/2):
a:= n-> denom(coeff(series(gf, x, n+3), x, n)):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 02 2013

Series[(x/(Exp[x]-1))^(3/2)*Exp[x/2], {x, 0, 25}] // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Mar 18 2014 *)

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).

A242225 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).

Original entry on oeis.org

1, 4, 48, 64, 1280, 3072, 86016, 49152, 2949120, 1310720, 11534336, 4194304, 1526726656, 2348810240, 12079595520, 3221225472, 73014444032, 51539607552, 137095356088320, 5772436045824, 3809807790243840, 725677674332160, 2023101395107840, 3166593487994880

Offset: 0

Jitender Singh, May 08 2014

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".
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

			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.

David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.

Cf. A241885.
Cf. also A222411/A222412, A350194/A350154.
Cf. A370416/A370417.

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);

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

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=2.
This formula is applicable for any rational root of an arithmetic function with respect to the Cauchy type product.

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

A350154 a(n) = denominator(k^n * [x^(2n+1)] sqrt(k)arccos(exp(-x^2/(2*k)))) for n >= 0 and fixed k > 0.

Original entry on oeis.org

1, 12, 480, 2688, 92160, 4055040, 805109760, 148635648, 2021444812800, 9037047398400, 41855798476800, 85571854663680, 1218840851644416000, 131634811977596928000, 30539276378802487296000, 26116346696355230515200, 72745993870031978496000, 8332722934203662991360000

Offset: 0

Robert B Fowler, Dec 16 2021

Denominators of a power series characterizing how powers of the cosine function converge to the Gaussian function.
As the cosine function is raised to increasing powers k, it converges to the Gaussian normal function. Let x be the standard deviation argument of the Gaussian function, and define a suitably scaled cosine function.
    G(x) = exp(-x^2/2),             Gaussian function.
    C(x,k) = (cos(x/sqrt(k)))^k,    k-th power of cosine function
    C(x,k) - G(x) = -x^4/(12k) + x^6/(24k) - x^6/(45x^2) + ...
The usefulness of this approximation lies within the "principal half-period" of C(x,k), defined as h_k = {x : abs(x) < sqrt(k)*Pi/2}. Within h_k, k can be any real number and C(x,k) is a good approximation to G(x) even for small k, although convergence to G(x) is only reciprocal in k. Outside h_k, negative cosine values occur and the approximation deteriorates.
If we define x(k) such that G(x) = C(x(k),k) then
    x = lim_{k->infinity} x(k).
The value of x(k) can be expressed as a polynomial in integer powers of x and k and coefficients A350194(n)/a(n), and characterizes how closely cosine powers approximate and converge to the Gaussian function.

			x(k) = x - (1/12)*(x^3/k) + (1/480)*(x^5/k^2) + (1/2688)*(x^7/k^3) - (1/92160)*(x^9/k^4) - (19/4055040)*(x^11/k^5) + (79/805109760)*(x^13/k^6) ...

David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.

Cf. A350194, A222411/A222412, A241885/A242225.

gf := sqrt(k)*arccos(exp(-x^2/(2*k))): assume(k > 0): assume(x > 0):
ser := series(gf, x, 80): seq(denom(k^n*coeff(ser, x, 2*n+1)), n=0..17); # Peter Luschny, Dec 19 2021

The definitions of G(x) and C(x,k) lead directly to the equation
  x(k) = sqrt(k)*arccos(exp(-x^2/(2k))),
which can be expanded into the power series
  x(k) = Sum_{n>=0} (x^(2n+1)/k^n) * (A350194(n)/a(n)).
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).

A350194 Numerators of a power series characterizing how powers of the cosine function converge to the Gaussian function.

Original entry on oeis.org

1, -1, 1, 1, -1, -19, 79, 11, -2339, -11813, 677, 2117, -308963, -64604977, 131301607, 263101079, -5614643, -1768132943, 46949081169401, 9606907803497, -10635113572583999, -158812278992229461, 8131167478793551, 9112944418860287, -40395223967437706149

Offset: 0

Robert B Fowler, Dec 19 2021

See A350154 for the denominators of this sequence of rational coefficients, as well as relevant comments, formulae, and examples.

David Broadhurst, Relations between A241885/A242225, A222411/A222412, and A350194/A350154.

Cf. A350154.

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).

A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

Original entry on oeis.org

1, -1, 1, 5, -21, -399, 869, 39325, -334477, -28717403, 59697183, 8400372435, -34429291905, -7199255611995, 14631594576045, 4251206967062925, -68787420596367165, -26475975382085110035, 53392138323683746235, 26275374869163335461975, -105772979046693606062363

Offset: 1

Eric W. Weisstein, Aug 20 2008

			1/sqrt(x^(-1)) - sqrt(x^(-1))/8 + (x^(-1))^(3/2)/128 + (5*(x^(-1))^(5/2))/1024 - (21*(x^(-1))^(7/2))/32768 + ...

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. [Except for the signs, see the unnumbered table on p. 7.]
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. [Except for the signs, see Table 4.]
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A061549, A088802 (denominators), A222411, A222412.

H := proc(n) local S, i; S := (x/(exp(x)-1))^(3/2)*exp(x/2);
-pochhammer(1/2,n-1)*coeff(series(S,x,n+2),x,n)*2^(4*n-1-add(i,i= convert(n,base,2))) end:
A143503 := n -> (-1)^irem(n-1,6)*H(n-1);
seq(A143503(n), n=1..16); # Peter Luschny, Apr 05 2014

Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* Vaclav Kotesovec, Oct 09 2023 *)

More terms from Vaclav Kotesovec, Oct 09 2023

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A222412 Denominators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A242225 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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A350154 a(n) = denominator(k^n * [x^(2*n+1)] sqrt(k)*arccos(exp(-x^2/(2*k)))) for n >= 0 and fixed k > 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A350194 Numerators of a power series characterizing how powers of the cosine function converge to the Gaussian function.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A350154 a(n) = denominator(k^n * [x^(2n+1)] sqrt(k)arccos(exp(-x^2/(2*k)))) for n >= 0 and fixed k > 0.