A242225 -id:A242225 - 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

A222411 Numerators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

Original entry on oeis.org

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

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. A222412 (denominators).

Cf. also A241885/A242225, A350194/A350154.

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

Series[(x/(Exp[x]-1))^(3/2)*Exp[x/2], {x, 0, 25}] // CoefficientList[#, x]& // Numerator (* 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).

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).

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).

A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 1, 1, -3, 1, -3, 1, 1, -1, 1, -19, -3, 5, 5, -5, 1, 79, -19, -9, 5, 5, -3, 1, 275, 79, -133, -21, 35, 7, -7, 1, -2339, 275, 79, -133, -21, 7, 7, -2, 1, -11813, -2339, 825, 79, -399, -189, 21, 3, -9, 1, 14217, -11813, -2339, 1375, 395, -399, -63, 15, 15, -5, 1

Offset: 0

Peter Luschny, Feb 18 2024

Nørlund polynomials N(a, n, x) are generalizations of the powers 1, x, x^2, ... as well as of the Bernoulli polynomials 1, x - 1/2, x^2 - x + 1/6, ...
Parameter a = 0 gives the first case and a = 1 the second case. Here, we consider the case a = 1/2. You can think of it as a kind of square root of the Bernoulli polynomials. We give the coefficients of these polynomials, this sequence for the numerators, and A370415 for the denominators.
We also give the values of these polynomials at the point x = 1, which are analogous to the Bernoulli numbers; A370416 for the numerators, and A370417 for the denominators.

			The lists of rational coefficients start:
  [0] [        1]
  [1] [     -1/4,        1]
  [2] [     1/48,     -1/2,         1]
  [3] [     1/64,     1/16,      -3/4,       1]
  [4] [  -3/1280,     1/16,       1/8,      -1,     1]
  [5] [ -19/3072,   -3/256,      5/32,    5/24,  -5/4,    1]
  [6] [ 79/86016,  -19/512,    -9/256,    5/16,  5/16, -3/2,    1]
  [7] [275/49152, 79/12288, -133/1024, -21/256, 35/64, 7/16, -7/4, 1]

Niels Erik Nørlund, Vorlesungen über Differenzenrechnung, Springer 1924.

Cf. A370415, A370416/A370417 (x=1), A241885/A242225 (x=0).

egf := (t/(exp(t) - 1))^(1/2)*exp(z*t):
ser := series(egf, t, 16): ct := n -> n!*coeff(ser, t, n):
seq(seq(numer(coeff(ct(n), z, k)), k = 0..n), n = 0..10);

Table[Numerator@CoefficientList[NorlundB[n, 1/2, x], x] , {n, 0, 10}] // Flatten

T(n, k) = numerator( n! * [z^k] [t^n] (t / (exp(t) - 1))^(1/2)*exp(z*t) ).

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).

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A222411 Numerators in Taylor series expansion of (x/(exp(x) - 1))^(3/2)*exp(x/2).

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

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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.

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

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A370414 T(n, k) = numerator([x^n] N(1/2, n, x)) where N(a, n, x) is the n-th Nørlund polynomial.

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