A241885 -id:A241885 - cp's OEIS frontend

A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

1, 1, 7, 139, 5473, 357721, 34988647, 4784061619, 871335013633, 203906055033841, 59618325600871687, 21297483077038703899, 9127322584507530151393, 4621897483978366951337161, 2730069675607609356178641127, 1860452328661957054823447670979, 1448802510679254790311316267306753

Offset: 0

Paul D. Hanna, Dec 20 2006

nonn

Previous name was: Column 0 and row sums of symmetric triangle A126155.

This is the square root of the Euler numbers (A122045) with respect to the Cauchy type product as described by J. Singh (see link and the second Maple program) normalized by 2^n. A241885 shows the corresponding sqrt of the Bernoulli numbers. - Peter Luschny, May 07 2014

			E.g.f.: A(x) = 1 + x^2/2! + 7*x^4/4! + 139*x^6/6! + 5473*x^8/8! + 357721*x^10/10! + ...
where the logarithm begins:
log(A(x)) = x^2/2! + 4*x^4/4! + 64*x^6/6! + 2176*x^8/8! + 126976*x^10/10! + 11321344*x^12/12! + ...
compare the logarithm to
A(x)^4 = 1 + 4*x^2/2! + 64*x^4/4! + 2176*x^6/6! + 126976*x^8/8! + 11321344*x^10/10! + ...

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 366.

G. C. Greubel, Table of n, a(n) for n = 0..200
Peter Bala, A triangle for calculating A126156
Alain Connes, Caterina Consani and Henri Moscovici, Zeta zeros and prolate wave operators, arXiv:2310.18423 [math.NT], Oct 2023, p.31.
Denis S. Grebenkov, Vittoria Sposini, Ralf Metzler, Gleb Oshanin, and Flavio Seno, Exact distributions of the maximum and range of random diffusivity processes, New J. Phys. (2021) Vol. 23, 023014.
Jitender Singh, On an arithmetic convolution, arXiv:1402.0065 [math.NT], 2014.

Diagonals: A126157, A126158.

Cf. A126155, A365672.

A126156 := proc(n)
        sqrt(sec(sqrt(2)*z)) ;
        coeftayl(%,z=0,2*n) ;
        %*(2*n)! ;
end;
seq(A126156(n),n=0..10) ; # Sergei N. Gladkovskii, Oct 31 2011
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 -> (-2)^n*g(euler, 2*n);
seq(a(n), n=0..14); # Peter Luschny, May 07 2014
# Alternative: an algorithm as described by Peter Bala, see also A365672:
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then
T(n, k-1) else (n - k + 1) * (2 * (n - k) + 1) * T(n, k - 1) + T(n - 1, k)
fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..14);  # Peter Luschny, Sep 29 2023

a[n_] := SeriesCoefficient[ Sqrt[ Sec[ Sqrt[2]*x]], {x, 0, 2 n}]*(2*n)!; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Nov 29 2013, after Sergei N. Gladkovskii *)

a(n):=if n=0 then 1 else 1/(4*n)*sum(binomial(2*n,2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*bern(2*k)*a(n-k)),k,1,n); /* Vladimir Kruchinin, Feb 25 2015 */

a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(2*n,2*k)*(k/(2*n)-1)*(-2)^k,k,1,n);
makelist(a[n],n,0,30); /* Tani Akinari, Sep 11 2023 */

/* E.g.f. A(x) = exp( Integral^2 A(x)^4 dx^2 ): */
{a(n)=local(A=1+x*O(x)); for(i=1, n, A=exp(intformal(intformal(A^4 + x*O(x^(2*n))))) ); (2*n)!*polcoeff(A, 2*n, x)}
for(n=0,20,print1(a(n),", "))

{a(n) = local(A=1+x); for(i=1,n, A = exp( intformal( A^2 * intformal( 1/A^2 + x*O(x^n)) ) ) ); n!*polcoeff(A,n)}
for(n=0,20,print1(a(2*n),", "))

{a(n)=-(n<1)-sum(j=0,n,sum(k=0,j/2,(2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1))))}; /* Tani Akinari, Sep 28 2023 */

def A126156(n): return A126155(n, 0)
print([A126156(n) for n in range(17)])  # Peter Luschny, Dec 14 2023

a(n) = Sum_{k=0..n} A087736(n,k)*3^(n-k). - Philippe Deléham, Jul 17 2007
E.g.f.: Sum_{n>=0} a(n)*x^(2*n)/(2*n)! = sqrt(sec(sqrt(2)*x)). - David Callan, Jan 03 2011
E.g.f. satisfies: A(x) = exp( Integral Integral A(x)^4 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, May 30 2015
E.g.f. satisfies: A(x) = exp( Integral A(x)^2 * Integral 1/A(x)^2 dx dx ), where A(x) = Sum_{n>=0} a(n)*x^(2*n)/(2*n)! and the constant of integration is zero. - Paul D. Hanna, Jun 02 2015
G.f.: 1/(1-x/(1-6*x/(1-15*x/(1-28*x/(1-45*x/(1-66*x/(1-91*x/(1-... or 1/U(0) where U(k) = 1-x*(k+1)*(2*k+1)/U(k+1); (continued fraction). [See Wall.] - Sergei N. Gladkovskii, Oct 31 2011
G.f.: 1/U(0) where U(k) = 1 - (4*k+1)*(4*k+2)*x/(2 - (4*k+3)*(4*k+4)*x/ U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 24 2012
G.f.: 1/G(0) where G(k) = 1 -x*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013
G.f.: Q(0), where Q(k) = 1 - x*(2*k+1)*(k+1)/( x*(2*k+1)*(k+1) - 1/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Oct 09 2013
a(n) ~ 2^(5*n+2) * n^(2*n) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Jul 13 2014
a(n) = (1/(4*n))*Sum_{k=1..n} binomial(2*n,2*k)*((2^(2*k)-1)*2^(3*k)*(-1)^((k-1))*Bernoulli(2*k)*a(n-k)), a(0)=1. - Vladimir Kruchinin, Feb 25 2015
a(n) = Sum_{k=1..n} a(n-k)*binomial(2*n,2*k)*(k/(2*n)-1)*(-2)^k, a(0)=1. - Tani Akinari, Sep 11 2023
For n > 0, a(n) = -Sum_{j=0..n} Sum_{k=0..floor(j/2)} (2*n+1)!*(2*k-j)^(2*n)/(n!*(2*j+1)*(n-j)!*k!*(j-k)!*(-2)^(n+j-1)). - Tani Akinari, Sep 28 2023

New name based on a comment of David Callan, Peter Luschny, May 07 2014

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).

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).

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) ).

A242233 2^n*(C_n)^(1/2) in the Cauchy type product where C_n is the n-th Catalan number.

Original entry on oeis.org

1, 1, 3, 11, 41, 137, 347, 611, 5777, 98321, 677363, -4192197, -134908871, -617972327, 22749265099, 449951818387, -632325203423, -163681108703199, -2324079456844573, 33233931805782635, 1734259111955765577, 14135975420529458857, -777499293367428199109

Offset: 0

Peter Luschny, May 08 2014

sign

Cf. A000108, A126156, A241885.

f := sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)));
seq(2^n*n!*coeff(series(f,x,n+1),x,n),n=0..22);
# Second program with function g from A241885:
catalan := n -> binomial(2*n,n)/(n+1);
a := n -> 2^n*g(catalan, n); seq(a(n), n=0..22);

g[n_] := g[n] = (CatalanNumber[n] - Sum[Binomial[n, m] g[m] g[n - m], {m, 1, n - 1}])/2;
a[0] = 1; a[n_] := 2^n g[n];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Aug 02 2019, from 2nd Maple program *)

a[n]:=if n=0 then 1 else sum(a[n-k]*binomial(n, k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!), k, 1, n); makelist(a[n],n,0,50); /* Tani Akinari, Nov 05 2024 */

a(n) = 2^n*n!*[x^n](sqrt(exp(2*x)*(BesselI(0,2*x)-BesselI(1,2*x)))), where [x^n](f(x)) the coefficient of x^n in f(x).

For n > 0, a(n) = Sum_{k=1..n} a(n-k)*binomial(n,k)*(2*k)!*(3*k/(2*n)-1)*2^k/(k!*(k+1)!). - Tani Akinari, Nov 05 2024

cp's OEIS Frontend

A126156 Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.

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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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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).

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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.