A222412 -id:A222412

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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).

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

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

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.

FORMULA

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)/(m*g(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

EXAMPLE

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.

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 *)

CROSSREFS

Cf. A242225, A126156, A242233.

Cf. also A222411/A222412, A350194/A350154.

Cf. A370416/A370417.

KEYWORD

sign,frac

AUTHOR

Jitender Singh, May 01 2014

EXTENSIONS

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

STATUS

approved

A222411

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

+10
7

1, -1, -1, 5, 7, -19, -869, 715, 2339, -200821, -12863, 2117, 7106149, -64604977, -131301607, 7629931291, 174053933, -19449462373, -46949081169401, 355455588729389, 10635113572583999, -6511303438681407901, -349640201588122693, 9112944418860287

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

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.

FORMULA

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).

EXAMPLE

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

MAPLE

gf:= (x/(exp(x)-1))^(3/2)*exp(x/2):

a:= n-> numer(coeff(series(gf, x, n+3), x, n)):

seq(a(n), n=0..25); # Alois P. Heinz, Mar 02 2013

MATHEMATICA

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

CROSSREFS

Cf. A222412 (denominators).

Cf. also A241885/A242225, A350194/A350154.

KEYWORD

sign,frac

AUTHOR

N. J. A. Sloane, Feb 28 2013

STATUS

approved

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).

+10
7

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

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.

FORMULA

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.

EXAMPLE

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.

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 -> denom(g(bernoulli, n));

seq(a(n), n=0..23);

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[Denominator[Factor[g[k]]], {k, 0, 15}] // TableForm

(* Alternative: *)

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

CROSSREFS

Cf. A241885.

Cf. also A222411/A222412, A350194/A350154.

Cf. A370416/A370417.

KEYWORD

nonn,frac

AUTHOR

Jitender Singh, May 08 2014

EXTENSIONS

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

STATUS

approved

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.

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..17.

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

FORMULA

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).

EXAMPLE

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) ...

MAPLE

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

CROSSREFS

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

KEYWORD

frac,nonn

AUTHOR

Robert B Fowler, Dec 16 2021

STATUS

approved

A350194

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

+10
6

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

COMMENTS

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

LINKS

Table of n, a(n) for n=0..24.

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

FORMULA

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).

CROSSREFS

Cf. A350154.

KEYWORD

frac,sign

AUTHOR

Robert B Fowler, Dec 19 2021.

STATUS

approved

A143503

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..21.

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.

EXAMPLE

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 + ...

MAPLE

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

MATHEMATICA

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

CROSSREFS

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

KEYWORD

sign,frac

AUTHOR

Eric W. Weisstein, Aug 20 2008

EXTENSIONS

More terms from Vaclav Kotesovec, Oct 09 2023

STATUS

approved

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