A046161 -id:A046161 - cp's OEIS frontend

A244420 Numerators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k, x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev polynomials of the first kind (type T).

1, 3, 1, 5, 5, 1, 35, 21, 7, 1, 63, 21, 9, 9, 1, 231, 165, 165, 55, 11, 1, 429, 1287, 715, 143, 39, 13, 1, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 12155, 2431, 1547, 1547, 595, 85, 17, 17, 1, 46189, 37791, 12597, 6783, 2907, 969, 969, 171, 19, 1, 88179, 146965, 101745, 14535, 6783, 20349, 5985, 665, 105, 21, 1

Offset: 0

Wolfdieter Lang, Aug 04 2014

This expansion is due to the Riordan property of the triangle A084930. The inverse of the lower triangular matrix built by A084930 is therefore also a (rational) Riordan triangle, namely ((2 - c(z/4))/(1-z), -1 + c(z/4)) in the standard notation, where c is the o.g.f. of A000108 (Catalan).
For the denominators of this triangle see A244421.
The expansion is x^n = sum(R(n,m)*Todd(m, x), m=0..n), n >= 0, with the rational triangle with entries R(n,m) = a(n, m)/b(n, m) with b(n, m) = A244421(n, m).
If one uses instead the expansion of (4*x)^n one finds the integer triangle A111418: (4*x)^n = sum(A111418(n,k) * Todd(k, x), k=0..n).
The row sums of the rational triangle R(n,m) are identically 1. The alternating row sums have o.g.f. 1/sqrt(1-x) which generates A001790(n)/A046161(n) (see a Michael Somos comment on A046161), namely 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, ...
From Wolfdieter Lang, Jun 13 2016: (Start)
R(n,m) = a(n, m)/ A244421(n, m) is also the rational triangle for the expansion cos^{2*n+1}(x) = Sum_{m=0..n} R(n, m)*cos((2*m+1)*x), n >= 0, m = 0..n. Compare with the odd numbered rows of A273496. In terms of Chebyshev T-polynomials (A053120) this is the identity x^(2*n+1) = Sum_{m=0..n} R(n, m)*T(2*m+1, x).
S(n,m) = (-1)^m*a(n, m)/ A244421(n, m) is the rational triangle for the expansion sin^{2*n+1}(x) = Sum_{m=0..n} S(n, m)*sin((2*m+1)*x), n >= 0, m = 0..n. In terms of Chebyshev S-polynomials (A049310) this is equivalent to the identity (4 - x^2)*n = Sum_{m=0..n} (-1)^m * binomial(n, n-m)*S(2*m,x), n >= 0.
(End)

			The numerator triangle a(n,m) begins:
  n\m      0      1      2     3    4     5    6   7   8   9
  0:       1
  1:       3      1
  2:       5      5      1
  3:      35     21      7     1
  4:      63     21      9     9    1
  5:     231    165    165    55   11     1
  6:     429   1287    715   143   39    13    1
  7:    6435   5005   3003  1365  455   105   15   1
  8:   12155   2431   1547  1547  595    85   17  17   1
  9:   46189  37791  12597  6783 2907   969  969 171  19   1
  ...
The rational triangle R(n,m) begins:
  n\m       0        1         2       3        4        5
  0:        1
  1:      3/4      1/4
  2:      5/8     5/16      1/16
  3:    35/64    21/64      7/64    1/64
  4:   63/128    21/64      9/64   9/256    1/256
  5:  231/512  165/512  165/1024 55/1024  11/1024   1/1024
  ...
The next rows are:
  n=6: 429/1024, 1287/4096, 715/4096, 143/2048, 39/2048, 13/4096, 1/4096,
  n=7: 6435/16384, 5005/16384, 3003/16384, 1365/16384, 455/16384, 105/16384, 15/16384, 1/16384,
  n=8: 12155/32768, 2431/8192, 1547/8192, 1547/16384, 595/16384, 85/8192, 17/8192, 17/65536, 1/65536,
  n=9: 46189/131072, 37791/131072, 12597/65536, 6783/65536, 2907/65536, 969/65536, 969/262144, 171/262144, 19/262144, 1/262144,
  ...
Expansions:
x^2 = 5/8 * Todd(0,x) + 5/16 * Todd(1,x) + 1/16 * Todd(2,x) = 5/8 + (5/16)*(-3 + 4*x) +(1/16)*(5 -20*x + 16*x^2).
x^3 = (35*Todd(0, x) + 21*Todd(1, x) + 7*Todd(2, x) + 1*Todd(3, x))/64 = (35 + 21*(-3+4*x) + 7*( 5-20*x+16*x^2) + (-7+56*x-112*x^2+64*x^3))/64.
For the Todd polynomials see the coefficient table A084930.

Wolfdieter Lang, First rows of the triangles.

Cf. A084930, A244421, A000108, A001790, A046161, A111418, A273496.

a(n, m) = numerator(R(n, m)) with the rationals Riordan matrix elements R(n, m)= [x^m]R(n, x), with the row polynomials R(n, x) generated by ((2 - c(z/4))/(1-z))/(1 - x*(-1 + c(z/4))) = 2*((1+x)*(z-1) + (1-x)*sqrt(1-z))/((1-z)*((1+x)^2*z - 4*x)), where c(x) is the o.g.f. of the Catalan numbers A000108.

The rationals R(n, m) = binomial(2*n+1, m)/2^(2*n). - Wolfdieter Lang, Jun 12 2016

A276738 Irregular triangle read by rows: T(n,m) = coefficients in a power/Fourier series expansion of an arbitrary anharmonic oscillator's exact phase space trajectory.

Original entry on oeis.org

-1, -1, 5, -1, 12, -32, -1, 14, 7, -126, 231, -1, 16, 16, -160, -160, 1280, -1792, -1, 18, 18, -198, 9, -396, 1716, -66, 2574, -12870, 14586, -1, 20, 20, -240, 20, -480, 2240, -240, -240, 6720, -17920, 2240, -35840, 129024, -122880, -1, 22, 22, -286, 22, -572, 2860, 11, -572, -286, 8580, -24310, -286, 4290, 8580, -97240, 184756, 715

Offset: 1

Bradley Klee, Sep 16 2016

Irregular triangle read by rows (see examples). Consider an arbitrary anharmonic oscillator with Hamiltonian energy: H=(1/2)*b^2=(1/2)*(p^2+q^2) + Sum_{i=3} 2*v_i*q^i, and a stable minimum at (p,q)=(0,0). The phase space trajectory can be written in polar phase space coordinates as (q,p) = (R(x)cos(x),R(x)sin(x))=(R(Q)Q,R(Q)P). The present triangle determines a power / Fourier series of R(Q): R(Q) = b * (1 + sum b^n*T(n,m)*f(n,m) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...A000041(n). The basis functions f(n,m) are constructed from partitions of "n" listed in reverse lexicographic order. Partition n=(z_1+z_2+...z_j) becomes 2*Q^((z_1+2)+(z_2+2)+...(z_j+2))*v_{z_1+2}*v_{z_2+2}*...*v_{z_j+2} (see examples). This sequence transforms into A273506/A273507 by setting v_i=0 for odd i, v_i:=(-1)^(i/2-1)/2/(i!) otherwise, and (1/2)*b^2 = 2*k. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n/m  1    2     3     4     5     6      7
--------------------------------------------
1  | -1
2  | -1   5
3  | -1   12   -32
4  | -1   14    7   -126   231
5  | -1   16    16  -160  -160   1280  -1792
--------------------------------------------
R[1,Q] = -2*v_3*Q^3
R[2,Q] = -2*v_4*Q^4 + 10*v_3^2*Q^6
R[Q]   = b*(1+b*(-2*v_3*Q^3)+b^2*(-2*v_4*Q^4 + 10*v_3^2*Q^6 ))+O(b^4)
Construct basis for R[4,Q]; List partitions: {{4}, {3, 1}, {2, 2}, {2, 1, 1}, {1, 1, 1, 1}}; Transform Plus 2: {{v_6}, {v_5, v_3}, {v_4, v_4}, {v_4, v_3, v_3}, {v_3, v_3, v_3, v_3}}; Multiply: {v_6, v_5*v_3, v_4^2, v_4*v_3^2, v_3^4}; don't forget power of Q and factor of 2: {2*v_6*Q^6, 2*v_5*v_3*Q^8, 2*v_4^2*Q^8, 2*v_4*v_3^2*Q^10, 2*v_3^4*Q^12}.

Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.

Arbitrary Oscillator: A276814, A276815, A276816, A276817.

Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496.

R[n_] := b Plus[1, Total[b^# R[#, q] & /@ Range[n]]]
Vp[n_] := Total[2 v[# + 2] q^(# + 2) & /@ Range[n]]
H[n_] := Expand[1/2*r^2 + Vp[n]]
RRules[n_] :=  With[{H = Series[ReplaceAll[H[n], {q -> R[n] Q, r -> R[n]}], {b, 0, n + 2}]},  Function[{rules},
    Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]]
basis[n_] :=  Times[Times @@ (v /@ #), Q^Total[#],2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
TriangleRow[n_, rules_] := With[{term = Expand[rules[[n, 2]]]},
  Coefficient[term, #] & /@ basis[n]]
With[{rules = RRules[10]}, TriangleRow[#, rules] & /@ Range[10]]

A276816 Irregular triangle read by rows: T(n,m) = coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact period.

Original entry on oeis.org

-24, 480, -120, 6720, 3360, -241920, 1774080, -560, 40320, 40320, -1774080, 20160, -3548160, 61501440, -591360, 92252160, -1845043200, 8364195840, -2520, 221760, 221760, -11531520, 221760, -23063040, 461260800, 110880, -23063040, -11531520, 1383782400, -15682867200, -11531520, 691891200, 1383782400, -62731468800, 476759162880

Offset: 1

Bradley Klee, Sep 18 2016

The phase space trajectory A276738 has phase space angular velocity A276814 and differential time dependence A276815. We calculate the period K = Int dt over the range [2*Pi, 0], trivial to compute from A276815 using A273496. Then K/(2*Pi) = 1 + sum b^(2n)*T(n,m)*f'(n,m); where the sum runs over n = 1, 2, 3 ... and m = 1, 2, 3, ... A000041(2n), and f'(n,m) = f(2n,m) of A276738 with Q=1/2. Choosing one point from the infinite dimensional coefficient space--v_i=0 for odd i, v_i=(-1)^(i/2-1)/2/(i!) otherwise--setting b^2 = 4*k, and summing over the entire table obtains the EllipticK expansion 2*A038534/A038533. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n/m   1     2     3         4         5
------------------------------------------
1  | -24   480
2  | -120  6720  3360   -241920   1774080
------------------------------------------
For pendulum values, f'(1,*)={(-1/384), 0}, f'(2,*) = {1/46080, 0, 1/294912, 0, 0}. Then K/(2Pi) = 1+(-1/384)*(-24)*4*k+((1/46080)*(-120)+(1/294912)*3360)*16*k^2=1+(1/4)*k + (9/64)*k^2, the first few terms of EllipticK.

Bradley Klee, Table of n, a(n) for n = 1..1537
Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.
Bradley Klee, A period function for anharmonic oscillations, Wolfram Community, 2016.

Arbitrary Oscillator: A276738, A276814, A276815, A276817.
Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496.
EllipticK: A038534, A038533.
Partition Triangles: A263633, A080577, A036038, A036040, A080575, A178867.

RExp[n_]:=Expand[b Plus[R[0], Total[b^# R[#] & /@ Range[n]]]]
RCalc[n_]:=With[{basis =Subtract[Tally[Join[Range[n + 2], #]][[All, 2]],Table[1, {n + 2}]] & /@ IntegerPartitions[n + 2][[3 ;; -1]]},
Total@ReplaceAll[Times[-2, Multinomial @@ #, v[Total[#]],Times @@ Power[RSet[# - 1] & /@ Range[n + 2], #]] & /@ basis, {Q^2 -> 1, v[2] -> 1/4}]]
dt[n_] := With[{exp = Normal[Series[-1/(1 + x)/.x -> Total[(2 # v[#] RExp[n - 1]^(# - 2) &/@Range[3, n + 2])], {b, 0, n}]]},
Expand@ReplaceAll[Coefficient[exp, b, #] & /@ Range[n], R -> RSet]]
RingGens[n_] :=Times @@ (v /@ #) & /@ (IntegerPartitions[n]/. x_Integer :> x + 2)
tri[m_] := MapThread[Function[{a, b},Times[-# /. v[n_] :> Q^n /. Q^n_ :>  Binomial[n, n/2],(1/2) Coefficient[a, #]] & /@ b], {dt[2 m][[2 #]] & /@ Range[m], RingGens[2 #] & /@ Range[m]}]
RSet[0] = 1; Set[RSet[#], Expand@RCalc[#]] & /@ Range[2*7];
tri7 = tri[7]; tri7 // TableForm
PeriodExpansion[tri_, n_] := ReplaceAll[ 1 + Dot[MapThread[ Dot, {tri,
  2 RingGens[2 #] & /@ Range[n]}], (2 h)^(Range[n])], {v[m_] :> (v[m]*(1/2)^m)}]
{#,SameQ[Normal@Series[(2/Pi)*EllipticK[k],{k,0,7}],#]}&@ReplaceAll[
PeriodExpansion[tri7,7],{v[n_/;OddQ[n]]:>0,v[n_]:> (-1)^(n/2-1)/2/(n!),h->2 k}]

A276817 Irregular triangle read by rows: T(n,m) = coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact differential precession.

Original entry on oeis.org

-1, 2, 6, -3, -16, 8, -48, 4, 30, -20, 140, 10, -140, 420, -5, -48, 36, -288, -24, 384, -1280, 12, -192, -96, 1920, -3840, 6, 70, -56, 504, 42, -756, 2772, -28, 504, 252, -5544, 12012, 14, -252, -252, 2772, 2772, -24024, 36036, -7, -96, 80, -800, -64, 1280, -5120, 48, -960, -480, 11520, -26880, -32, 640, 640, -7680

Offset: 0

Bradley Klee, Sep 18 2016

Irregular triangle read by rows (see examples).

Consider an axially symmetric oscillator in two dimensions with polar coordinates ( r, y ). By conservation of angular momentum, replace the cyclic angle coordinate y with dy/dt = 1/r^2. The system becomes one-dimensional in r, with an effective potential including the 1/r^2 term. Assume that the effective potential has a minimum around r0 and apply a linear transform r --> q = r-r0. Radial oscillations around the effective potential minimum follow the exact solution of A276738, A276814, A276815, A276816. Now dy = dx (dy/dt) / (dx/dt) = dx * Sum b^n*T(n,m)*F(n,m), with n=1,2,3.... and m=1,2,3...A000070(n). Basis functions F(n,m) are an ordered union over A276738's f(n,m): F(n,m')={ (1/r0^2)*(Q/r0)^n } & Append_{i=1..n}_{m=1..A000041(n)} (1/2/r0^2)*(Q/r0)^(n - i)*f(i,m), where each successive term f(i,m) is appended such that index m' inherets the ordering of each m index (see examples). Integrating dx over a range of 2 Pi loses all odd rows, as in A276815 / A276816. This sequence is a useful tool in classical and relativistic astronomy (follow links to Wolfram demonstrations).

			n/m   1   2    3    4      5     6    7
------------------------------------------
0  | -1
1  |  2   6
2  | -3  -16   8   -48
3  |  4   30  -20   140   10   -140   420
------------------------------------------
Construction of F(2,_). List f(i,_) basis sets: {f(1,_)={2*Q^3*v_3},f(2,_)= {2*Q^4*v_4, 2*Q^6*v_3^2}}; Integrate and join: F(2,_)={(1/r0^2)*(Q/r0)^2,2*Q^3*v_3*(1/2/r0^2)*(Q/r0),2*Q^4*v_4*(1/2/r0^2), 2*Q^6*v_3^2*(1/2/r0^2)}={Q^2/r0^4,Q^4*v_3/r0^3,Q^4*v_4/r0^2,Q^6*v_3^2/r0^2}.
dy Expansion to second order: dy=dx(-(1/r0^2)+b^2*(2*Q/r0^3 + 6*Q^3*v_3/r0^2)+b^3*(-3*Q^2/r0^4 - 16*Q^4*v_3/r0^3 - 48*Q^6*v_3^2/r0^2 + 8*Q^4*v_4/r0^2)+O(b^3).
Cancellation of higher orders 1 to infinity and closed orbits. Kepler values {r0 = 1, v_n := ((n - 1)/4)*(-1)^n} yield dy = -dx. Harmonic oscillator values {r0 = Sqrt[2], v_n := ((-1)^n*(n + 1)/4/2)/sqrt[2]^n} yield dy = -(1/2)*dx. Parity symmetric conjectured values {r0=Sqrt[1/R],v_n odd n := 0,v_n even n := R^(n/2 - 1)*(n/8)} yield dy = -R*dx (see attached image "Pentagonal Orbits")?

R. M. Wald, General Relativity, University of Chicago press, 2010, pages 139-143.
J.A. Wheeler, A Journey into Gravity and Spacetime, Scientific American Library, 1990, pages 168-183.

Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.
Bradley Klee, Estimating Planetary Perihelion Precession, Wolfram Demonstrations Project, 2106.
Bradley Klee, Exact and Approximate Relativistic Corrections to the Orbital Precession of Mercury, Wolfram Demonstrations Project, 2016.
Bradley Klee, Pentagonal Orbits
Seqfans, Another planetary sequence, Seqfans mailing list, September 2016.

Arbitrary Oscillator: A276738, A276814, A276815, A276816.

Pendulum: A273506, A273507, A274076, A274078, A274130, A274131, A038534, A056982, A000984, A001790, A038533, A046161, A273496.

R[n_] := b Plus[1, Total[b^# R[#, q] & /@ Range[n]]]
Vp[n_] := Total[2 v[# + 2] q^(# + 2) & /@ Range[n]]
H[n_] := Expand[1/2*r^2 + Vp[n]]
RRules[n_] :=  With[{H = Series[ReplaceAll[H[n], {q -> R[n] Q, r -> R[n]}], {b, 0, n + 2}]},  Function[{rules},
    Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, q] -> Expand[-ReplaceAll[ Coefficient[H, b^(# + 2)], {R[#, q] -> 0}]] & /@ Range[n]]]]
xDot[n_] := Expand[Normal@Series[ReplaceAll[ Q^2 D[D[q[t], t]/q[t], t], {D[q[t], t] -> R[n] P, q[t] -> R[n] Q, r -> R[n], D[q[t], {t, 2}]
->  ReplaceAll[D[-(q^2/2 + Vp[n]), q], q -> R[n] Q]} ], {b, 0, n}] /. RRules[n] /. {P^2 -> 1 - Q^2}]
ydot[n__] := Expand[Normal@Series[1/(r0 + q)^2 /. {q -> R[n] Q} /. RRules[n], {b, 0, n}]]
dy[n_] := Expand@Normal@Series[ydot[n]/xDot[n], {b, 0, n}]
basis[n_] :=  Times[Times @@ (v /@ #), Q^Total[#],2] & /@ (IntegerPartitions[n] /. x_Integer :> x + 2)
extendedBasis[n_] :=Flatten[(1/2/r0^2) (Q/r0)^(n - #) basis[#] & /@ Range[0, n]]
TriangleRow[n_, func_] := Coefficient[func, b^n #] & /@ extendedBasis[n]
With[{dy5 = dy[5]}, TriangleRow[#, dy5] /. v[_] -> 0 & /@ Range[0, 5]]
(*Kepler Test*)TrigReduce[dy[5] /. {Q -> Cos[x]}] /. {r0 -> 1, Cos[] -> 0, v[n] :> ((n - 1)/4)*(-1)^n}
(*Harmonic Test*)TrigReduce[dy[5] /. {Q -> Cos[x]}] /. {Cos[] -> 0, v[n] :> ((-1)^n*(n + 1)/4/2)/Sqrt[2]^n, r0 -> Sqrt[2]}
(*Conjecture*)TrigReduce[dy[5] /. {Q -> Cos[x]}] /. {Cos[] -> 0, v[n /; OddQ[n]] :> 0, v[n_] :> RR^(n/2 - 1)*n/8, r0 -> Sqrt[1/RR]}

A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^neuler(2n)x^n/(2n)).

Original entry on oeis.org

1, 1, 11, 173, 22931, 1319183, 233526463, 29412432709, 39959591850371, 8797116290975003, 4872532317019728133, 1657631603843299234219, 2718086236621937756966743, 1321397724505770800453750299, 1503342018433974345747514544039

Offset: 0

Johannes W. Meijer and Joseph Abate, Jan 03 2017

This sequence is related in a peculiar way to A223067, a sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes. See A280443 for more information.

Sergey Khrushchev, Orthogonal Polynomials and Continued Fractions, From Euler's point of view, Corollary 4.26, p. 192, 2008.

Cf. A046161 (denominators).

Cf. A000364 (Euler numbers), A223067, A255881, A280443.

nmax:=14: f := series(exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1)), x=0, nmax+1): for n from 0 to nmax do a(n) := numer(coeff(f, x, n)) od: seq(a(n), n=0..nmax);

def A280442_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    def g(x): return exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2^(2*n - sum(n.digits(2)))
    return [d(n)*R[n] for n in (0..prec)]
print(A280442_list(14)) # Peter Luschny, Jan 05 2017

a(n) = numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * euler(2*n)*x^n/(2*n)).

Let S = Sum_{n>=0} (-1)^n*euler(2*n)*x^n/(2*n) and w(n) = A005187(n) then a(n) = 2^w(n) * [x^n] exp(S). - Peter Luschny, Jan 05 2017

A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).

Original entry on oeis.org

1, 2, 1, 4, 3, 16, 5, 32, 35, 256, 63, 512, 231, 2048, 429, 4096, 6435, 65536, 12155, 131072, 46189, 524288, 88179, 1048576, 676039, 8388608, 1300075, 16777216, 5014575, 67108864, 9694845, 134217728, 300540195

Offset: 0

N. J. A. Sloane

Also rational part of denominator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A004731).

			1, 1, 1/2*Pi, 2, 3/4*Pi, 8/3, 15/16*Pi, 16/5, 35/32*Pi, 128/35, 315/256*Pi, ...
The sequence Gamma(n/2+1)/Gamma(n/2+1/2), n >= 0, begins 1/Pi^(1/2), (1/2)*Pi^(1/2), 2/Pi^(1/2), (3/4)*Pi^(1/2), (8/3)/Pi^(1/2), (15/16)*Pi^(1/2), (16/5)/Pi^(1/2), ...

D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A004731.
Bisections are A001790 and A101926.
Cf. A004731, A046161, A001790, A001803, A101926.

if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1,k)/4^k else k := (n-1)/2; 4^k/binomial(2*k,k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));

Table[ Denominator[ Gamma[n/2+1]/Gamma[n/2+1/2]*Sqrt[Pi]^(1 - 2 Mod[n, 2])], {n, 0, 32}] (* Jean-François Alcover, Jul 16 2012 *)

A052468 Numerators in the Taylor series for arccosh(x) - log(2*x).

Original entry on oeis.org

1, 3, 5, 35, 63, 77, 429, 6435, 12155, 46189, 88179, 676039, 1300075, 5014575, 646323, 300540195, 583401555, 756261275, 4418157975, 6892326441, 22427411435, 263012370465, 514589420475, 2687300306925, 15801325804719, 61989816618513, 121683714103007

Offset: 1

Eric W. Weisstein

A055786 is the preferred version of this sequence.

			i*Pi/2 - arccosh(x) = i*x + (1/6)*i*x^3 + (3/40)*i*x^5 + (5/112)*i*x^7 + (35/1152)*i*x^9 + (63/2816)*i*x^11 + (231/13312)*i*x^13 + (143/10240)*i*x^15 + (6435/557056)*i*x^17 + ...
0, 1, 0, 1/6, 0, 3/40, 0, 5/112, 0, 35/1152, 0, 63/2816, 0, 231/13312, 0, 143/10240, 0, 6435/557056, 0, 12155/1245184, 0, 46189/5505024, 0, ... = A052468/A052469

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Secant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine

See A055786 for further information.
a(n)/A052469(n) = (1/(2*n))*A001790(n)/A046161(n) for n=>1.
Equals A162441(n+1)/(2n+1) for n=>1. - Johannes W. Meijer, Jul 06 2009

List([1..30], n-> NumeratorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019

[Numerator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017

a [n_]:=Numerator[(2 n - 1)! / (2^(2 n) n!^2)]; Array[a, 40] (* Vincenzo Librandi, Jul 10 2017 *)

{a(n) = numerator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019

[numerator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

a(n)/A052469(n) = A001147(n)/(A000165(n)*2*n). E.g., a(6) = 77 = 1*3*5*7*9*11 / gcd( 1*3*5*7*9*11, 2*4*6*8*10*12*12 ).
a(n) = numerator((2*n-1)!/(4^n * (n!)^2)). - Johannes W. Meijer, Jul 06 2009
Let z(n) = 2*(2*n+1)!*4^(-n-1)/((n+1)!)^2, then a(n) = numerator(z(n)), A162442(n) = denominator(z(n)), and z(n) = 1/(n+1) - Sum_{k=0..n}(-1)^k*binomial(n,k)*z(k). - Groux Roland, Jan 04 2011
a(n) = numerator(binomial(2n,n)/(n*2^(2n-1))). - Daniel Suteu, Oct 30 2017

Updated by Frank Ellermann, May 22 2011

Cross-references edited by Johannes W. Meijer, Jul 05 2009

A052469 Denominators in the Taylor series for arccosh(x) - log(2*x).

Original entry on oeis.org

4, 32, 96, 1024, 2560, 4096, 28672, 524288, 1179648, 5242880, 11534336, 100663296, 218103808, 939524096, 134217728, 68719476736, 146028888064, 206158430208, 1305670057984, 2199023255552, 7696581394432, 96757023244288, 202310139510784, 1125899906842624

Offset: 1

Eric W. Weisstein

			arccosh(x) = log(2x) - 1/(4*x^2) - 3/(32*x^4) - 5/(96*x^6) - ... for x>1.

Bronstein-Semendjajew, sprawotchnik po matematikje, 6th Russian ed. 1956, ch. 4.2.6.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosecant
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Cosine
Eric Weisstein's World of Mathematics, Inverse Hyperbolic Sine

Cf. A002595.

List([1..30], n-> DenominatorRat( Factorial(2*n-1)/(4^n*(Factorial(n))^2) )) # G. C. Greubel, May 18 2019

[Denominator(Factorial(2*n-1)/( 2^(2*n)* Factorial(n)^2)): n in [1..30]]; // Vincenzo Librandi, Jul 10 2017

a[n_] := Denominator[(2*n-1)!/(2^(2*n)*n!^2)]; Array[a, 21] (* Jean-François Alcover, May 17 2017 *)

{a(n) = denominator((2*n-1)!/(4^n*(n!)^2))}; \\ G. C. Greubel, May 18 2019

[denominator(factorial(2*n-1)/(4^n*(factorial(n))^2)) for n in (1..30)] # G. C. Greubel, May 18 2019

A052468(n) / a(n) = A001147(n) / ( A000165(n) *2*n )
From Johannes W. Meijer, Jul 06 2009: (Start)
a(n) = denom((2*n-1)!/( 4^n * (n!)^2)).
Equals 2*A162442(n+1) for n >= 1.
A052468(n)/a(n) = (1/(2*n))*A001790(n)/A046161(n) for n>=1.
(End)

Updated by Frank Ellermann, May 22 2001

A067002 Numerator of Sum_{k=0..n} 2^(k-2n) binomial(2n-2k,n-k) * binomial(n+k,n).

Original entry on oeis.org

1, 3, 21, 77, 1155, 4389, 33649, 129789, 4023459, 15646785, 122044923, 477084699, 7474326951, 29322359577, 230389968105, 906200541213, 57090634096419, 225004263791769, 1775033636579511, 7006711723340175, 110706045228774765

Offset: 0

N. J. A. Sloane, Feb 16 2002

Numerator of e(0,n) (see the Maple line).

The generating function of the full fraction is (1-2*x)^(-3/4). - R. J. Mathar, Nov 06 2011

			1, 3/2, 21/8, 77/16, 1155/128, 4389/256, 33649/1024, 129789/2048, 4023459/32768, ... = A067002/A046161.

V. H. Moll. The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.

Denominators are in A046161.

Cf. A067001, A223549, A223550.

e := proc(l,m) local k; add(2^(k-2*m)*binomial(2*m-2*k,m-k)*binomial(m+k,m)*binomial(k,l),k=l..m); end;

Numerator[Table[Sum[2^(k-2n) Binomial[2n-2k,n-k]Binomial[n+k,n],{k,0,n}],{n,0,30}]] (* Harvey P. Dale, Oct 19 2012 *)

Numerator of 2^n*Gamma(n + 3/4)/(Gamma(3/4)*n!). - R. J. Mathar, Nov 06 2011
Numerator of integral_{x>0} 1/(x^4 + 1)^(n+1) / (Pi*sqrt(2)). - Jean-François Alcover, Apr 29 2013
From Petros Hadjicostas, May 23 2020: (Start)
If fr(n) = A067002(n)/A046161(n), then fr(n) = P_n(0), where P_n(x) is the Boros-Moll polynomial mentioned in A223549 and A223550 (and whose coefficients are the numbers e(l,n) = A067001(n,n-l)/2^(2*n) that are mentioned in the Maple line below with l = 0..n).
Recurrence for fr(n): 4*n*(n - 1)*fr(n) = 8*(2*n - 1)*(n - 1)*fr(n-1) - (16*(n-1)^2 - 1)*fr(n-2) for n >= 2 with fr(0) = 1 and fr(1) = 3/2. (End)

A069955 Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->oo} W(n) = 2/Pi); then a(n) = numerator(W(n)).

Original entry on oeis.org

1, 3, 45, 175, 11025, 43659, 693693, 2760615, 703956825, 2807136475, 44801898141, 178837328943, 11425718238025, 45635265151875, 729232910488125, 2913690606794775, 2980705490751054825, 11912508103174630875, 190453061649520333125, 761284675790187924375

Offset: 0

Benoit Cloitre, Apr 27 2002

Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n>=1} 4*n^2/(4*n^2-1). Numerators are in A056982.

Orin J. Farrell and Bertram Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Seiichi Manyama, Table of n, a(n) for n = 0..832
Boris Gourévitch, L'univers de Pi.

Not the same as A001902(n).
Cf. A056982 (denominators), A001790, A046161.
W(n)=(3/4)*(A120995(n)/A120994(n)), n>=1.

a[n_] := Numerator[Product[1 - 1/(4*k^2), {k, 1, n}]]; Array[a, 20, 0] (* Amiram Eldar, May 07 2025 *)

a(n) = numerator(prod(k=1, n, 1-1/(4*k^2))); \\ Michel Marcus, Oct 22 2016

a(n) = numerator(W(n)), where W(n) = (2*n)!*(2*n+1)!/((2^n)*n!)^4.
W(n) = (2*n+1)*(binomial(2*n,n)/2^(2*n))^2 = (2*n+1)*(A001790(n)/A046161(n))^2 in lowest terms.
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n+1/2,n). - Peter Luschny, Apr 08 2016

cp's OEIS Frontend

A244420 Numerators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k, x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev polynomials of the first kind (type T).

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A276738 Irregular triangle read by rows: T(n,m) = coefficients in a power/Fourier series expansion of an arbitrary anharmonic oscillator's exact phase space trajectory.

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A276816 Irregular triangle read by rows: T(n,m) = coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact period.

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A276817 Irregular triangle read by rows: T(n,m) = coefficients in power/Fourier series expansion of an arbitrary anharmonic oscillator's exact differential precession.

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A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n*euler(2*n)*x^n/(2*n)).

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A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).

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A052468 Numerators in the Taylor series for arccosh(x) - log(2*x).

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A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^neuler(2n)x^n/(2n)).

A067002 Numerator of Sum_{k=0..n} 2^(k-2n) binomial(2n-2k,n-k) * binomial(n+k,n).