A056982 -id:A056982 - cp's OEIS frontend

A260832 a(n) = numerator(Jtilde2(n)).

1, 3, 41, 147, 8649, 32307, 487889, 1856307, 454689481, 1748274987, 26989009929, 104482114467, 6488426222001, 25239009088827, 393449178700161, 1535897056631667, 1537112996582116041, 6016831929058214523, 94316599529950360769, 369994845516850143483, 23244865440911268112681

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde2(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(2), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..830
Takashi Ichinose and Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto and Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 1).
Ling Long, Robert Osburn and Holly Swisher, On a conjecture of Kimoto and Wakayama, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.

Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
seq(a(n), n = 0..20); # Peter Luschny, Dec 08 2022

Numerator[Table[Sum[ (-1)^k*Binomial[-1/2, k]^2*Binomial[n, k], {k, 0, n}], {n,0,50}]] (* G. C. Greubel, Feb 15 2017 *)

a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2,k)^2*binomial(n, k)));

a(n) = numerator(sum(k=0, n, binomial(2*k, k)*binomial(4*k, 2*k)* binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))) / (2^(4*n)* binomial(2*n,n)));

Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
And 4n^2*J2(n) - (8n^2-8n+3)*J2(n-1) + 4(n-1)^2*J2(n-2) = 0 with J2(0) = 3*zeta(2) and J2(1) = 9*zeta(2)/4.
Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
Jtilde2(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(4*k,2*k)*binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))/(2^(4*n)*binomial(2*n,n)).
From Andrey Zabolotskiy, Oct 04 2016 and Dec 08 2022: (Start)
Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
Jtilde2(n) = A143583(n) / 16^n. (End)
a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - Peter Luschny, Dec 08 2022

A038534 Numerators of coefficients of EllipticK/Pi.

Original entry on oeis.org

1, 1, 9, 25, 1225, 3969, 53361, 184041, 41409225, 147744025, 2133423721, 7775536041, 457028729521, 1690195005625, 25145962430625, 93990019574025, 90324408810638025, 340357374376418025, 5147380044581630625, 19520119892056100625, 1187604094232693162025

Offset: 0

Wouter Meeussen, revised Jan 03 2001

The denominators are given in A038533.
Also numerators in expansion of the hypergeometric series 2F1(1/2,1/2; 1; x).
This means numerators of the expansion coefficients of 2*K(k)/Pi = 2F1(1/2,1/2; 1; k^2) in powers of k^2, with K(k) the complete elliptic integral of the first kind. The denominators are given in A056982. The period T of the plane pendulum (mass m, length L, Earth's gravity g, energy E) is 4*sqrt(L/g)*K(sin(phi_0/2)) with cos(phi_0) = -E/(m*g*L) (maximal phi value). See the Landau - Lifschitz reference, p. 30. - Wolfdieter Lang, May 29 2016
It is easy and inexpensive to make a satisfactory precision measurement of a(1)/4, a(2)/64, and a(3)/256 using a pendulum rigged from a computer mouse. In "Digital Pendulum Data Analysis" (see links) amplitude vs. time data is transformed to period vs. sin(phi_0/2)^2 data, thus allowing extraction of expansion coefficients as fit parameters. - Bradley Klee, Dec 25 2016

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 91, Eq. 2.1.
L. D. Landau und E. M. Lifschitz, Mechanik, Akademie Verlag, Berlin, 1967, p. 30 (Exercise 1 in chapter III, paragraph 11.)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
B. Klee, Digital Pendulum Data Analysis: Output, Github, 2016.
David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, arXiv:2109.00027 [math.AG], 2021. See (1.2) p. 1.
G. N. Watson, A Note on Gamma Functions.Edinburgh Mathematical Notes, 42, 1959, pp 7-9.

Cf. A001790, A038533, A038535, A056040, A056982, A060632.

swing := proc(n) option remember; if n = 0 then 1 elif n mod 2 = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
sigma := n -> 2^(add(i, i = convert(iquo(n, 2), base, 2))):
a := n -> (swing(2*n)/sigma(2*n))^2; seq(a(n),n=0..20); # Peter Luschny, Aug 06 2014

Numerator@ CoefficientList[ Series[ EllipticK@x, {x, 0, 19}]/Pi, x] (* Robert G. Wilson v, Jul 19 2007 *)

a(n) = 2^(-2*w(n))*binomial(2*n,n)^2 with w(n) = A000120(n), the number of 1's in binary expansion of n.
a(n) = A001790(n)^2.
a(n) = (A056040(2*n)/A060632(2*n))^2. - Peter Luschny, Aug 06 2014
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n-1/2,n). - Peter Luschny, Apr 08 2016
a(n) = numerator(((2*n)!/(2^(2*n)*(n!)^2))^2). - Stefano Spezia, May 01 2025

A038533 Denominator of coefficients of both EllipticK/Pi and EllipticE/Pi.

Original entry on oeis.org

2, 8, 128, 512, 32768, 131072, 2097152, 8388608, 2147483648, 8589934592, 137438953472, 549755813888, 35184372088832, 140737488355328, 2251799813685248, 9007199254740992, 9223372036854775808, 36893488147419103232, 590295810358705651712, 2361183241434822606848

Offset: 0

Wouter Meeussen, revised Jan 03 2001

Denominators are powers of 2 since EllipticK(x) = Pi * Sum_{n>=0} 2^(-4*n-1) * binomial(2*n,n)^2 * x^n and EllipticE(x) = Pi * Sum_{n>=0} 2^(-4*n-1) (-1)^(2*n) * binomial(2*n,n)^2 /(-2*n+1) * x^n.

David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, Notices of the American Mathematical Society, Vol. 69, No. 6 (2022), pp. 914-929; arXiv preprint, arXiv:2109.00027 [math.AG], 2021. See eq. (1.2), p. 914.
Index to divisibility sequences.

Cf. A000120, A038534, A038535.

Equals 2*A056982(n).

a[n_] := 2^(4*n - 2*DigitCount[n, 2, 1] + 1); Array[a, 20, 0] (* Amiram Eldar, Aug 03 2023 *)

a(n)=my(s=n); while(n>>=1, s+=n); 2<<(2*s) \\ Charles R Greathouse IV, Apr 07 2012

a(n) = 2^(1+4*n-2*w(n)) with w(n) = A000120(n) = number of 1's in binary expansion of n.

A273506 T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.

Original entry on oeis.org

1, -1, 7, 1, -1, 11, -1, 319, -143, 715, 1, -26, 559, -221, 4199, -2, 139, -323, 6137, -2261, 52003, 1, -10897, 135983, -4199, 527459, -52003, 37145, -1, 15409, -317281, 21586489, -52877, 7429, -88711, 1964315, 1, -76, 269123, -100901, 274873, -8671, 227447, -227447, 39803225, -2, 466003, -213739, 522629, -59074189, 226061641, -10690009, 25701511, -42077695, 547010035

Offset: 1

Bradley Klee, May 23 2016

Triangle read by rows ( see examples ). The phase space trajectory of a simple pendulum can be written as (q,p) = (R(Q)cos(Q),R(Q)sin(Q)), with scaled, canonical coordinates q and p. The present triangle and A273507 determine a power / Fourier series of R(Q): R(Q) = sqrt(4 *k) * (1 + sum k^n * (A273506(n,m)/A273507(n,m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. The period of an oscillator can be computed by T(k) = dA/dE, where A is the phase area enclosed by the phase space trajectory of conserved, total energy E. As we choose expansion parameter "k" proportional to E, the series expansion of the complete elliptic integral of the first kind follows from T(k) with very little technical difficulty ( see examples and Mathematica function R2ToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

For some remarks on this pendulum problem and an alternative way to compute a(n,m) / A273507(n,m) using Lagrange inversion see the two W. Lang links. - Wolfdieter Lang, Jun 11 2016

			n/m  1    2     3     4
------------------------------
1  |  1
2  | -1,  7
3  |  1, -1,    11
4  | -1,  319, -143, 715
------------------------------
R2(Q) = sqrt(4 k) (1 + (1/6) cos(Q)^4 k +  (-(1/45) cos(Q)^6 + (7/72) cos(Q)^8) k^2)
R2(Q)^2 = 4 k + (4/3) cos(Q)^4 k^2 + ( -(8/45) cos(Q)^6 + (8/9) cos(Q)^8)k^3 + ...
I2 = (1/(2 Pi)) Int dQ (1/2)R2(Q)^2 = 2 k + (1/4) k^2 + (3/32) k^3 + ...
(2/Pi) K(k) ~ (1/2)d/dk(I2) = 1 + (1/4) k + (9/64) k^2 + ...
From _Wolfdieter Lang_, Jun 11 2016 (Start):
The rational triangle r(n,m) = a(n, m) / A273507(n,m) begins:
n\m   1          2          3         4   ...
1:   1/6
2: -1/45        7/72
3:  1/630      -1/30      11/144
4: -1/14175   319/56700 -143/3240  715/10368
... ,
row n = 5: 1/467775 -26/42525 559/45360 -221/3888 4199/62208,
row 6: -2/42567525 139/2910600 -323/145800 6137/272160 -2261/31104 52003/746496,
row 7: 1/1277025750 -10897/3831077250 135983/471517200 -4199/729000 527459/13996800 -52003/559872 37145/497664,
row 8:
-1/97692469875 15409/114932317500 -317281/10945935000 21586489/20207880000 -52877/4199040 7429/124416 -88711/746496 1964315/23887872.
... (End)

Bradley Klee, Plane Pendulum and Beyond by Phase Space Geometry, arXiv:1605.09102 [physics.class-ph], 2016.
Wolfdieter Lang, Remarks on this entry and A273507
Wolfdieter Lang, Expansions for phase space coordinates for the plane pendulum

Denominators: A273507. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
RCoefficients[n_] :=  With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
    Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
       Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
R2ToEllK[NMax_] := D[Expand[(2)^(-2) ReplaceAll[R[NMax], RRules[NMax]]^2] /. {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0},k]
Flatten[Numerator@RCoefficients[10]]
R2ToEllK[10]

A273507 T(n, m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact phase space trajectory.

Original entry on oeis.org

6, 45, 72, 630, 30, 144, 14175, 56700, 3240, 10368, 467775, 42525, 45360, 3888, 62208, 42567525, 2910600, 145800, 272160, 31104, 746496, 1277025750, 3831077250, 471517200, 729000, 13996800, 559872, 497664, 97692469875, 114932317500, 10945935000, 20207880000, 4199040, 124416, 746496, 23887872

Offset: 1

Bradley Klee, May 23 2016

Triangle read by rows ( see example ). The numerator triangle is A274076.

Comments of A273506 give a definition of the fraction triangle, which determines an arbitrary-precision solution to the simple pendulum equations of motion. For more details see "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n/m  1      2      3     4
------------------------------
1  | 6
2  | 45,    72
3  | 630,   30,    144
4  | 14175, 56700, 3240, 10368
------------------------------

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

Numerators: A273506. Time Dependence: A274076, A274078, A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[
          Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
RCoefficients[n_] :=  With[{Rn = ReplaceAll[R[n], RRules[n]]}, Function[{a},
    Coefficient[Coefficient[Rn/2/Sqrt[k], k^a],
       Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
Flatten[Denominator@RCoefficients[10]]

A274076 T(n, m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.

Original entry on oeis.org

-2, 2, -2, -4, 8, -20, 2, -58, 14, -70, -4, 16, -344, 112, -28, 4, -556, 1064, -152, 308, -308, -8, 10256, -3368, 4576, -6248, 2288, -1144, 2, -1622, 33398, -98794, 34606, -4862, 2002, -1430, -4, 6688, -187216, 140384, -1242904, 59488, -25168, 77792, -48620

Offset: 1

Bradley Klee, Jun 09 2016

Triangle read by rows ( see examples ). The denominators are given in A274078.

The rational triangle A273506 / A273507 gives the coefficients for an exact solution of the plane pendulum's phase space trajectory. Differential time dependence for this solution also adheres to the simple form of a triangular summation: dt = dQ(-1+ sum k^n * (T(n, m)/A274078(n, m)) * cos(Q)^(2(n+m)) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3...n. Expanding powers of cosine ( Cf. A273496 ) it is relatively easy to integrate dt ( cf. A274130 ). One period of motion takes Q through the range [ 0 , -2 pi]. Integrating dt over this domain gives another (Cf. A273506) calculation of the series expansion for Elliptic K ( see examples and Mathematica function dtToEllK ). For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			The triangle T(n, m) begins:
n/m  1    2     3     4
------------------------------
1  | -2
2  |  2, -2
3  | -4,  8,  -20
4  |  2, -58,  14,  -70
------------------------------
The rational triangle T(n, m) / A274078(n, m) begins:
n/m    1        2         3       4
------------------------------------------
1  | -2/3
2  |  2/15,   -2/3
3  | -4/315,   8/27,   -20/27
4  |  2/2835, -58/945,  14/27,  -70/81
------------------------------------------
dt2(Q) = dQ(-1 - (2/3) cos(Q)^4 k +  ((2/15) cos(Q)^6  - (2/3) cos(Q)^8) k^2 ) + ...
dt2(Q) = dQ(-1 - (1/4) k - (9/64) k^2 + cosine series ) + ...
(2/Pi) K(k) ~ I2 = (1/(2 Pi)) Int dt2(Q) =  1 + (1/4) k + (9/64) k^2+ ...

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

Denominators: A274078. Phase Space Trajectory: A273506, A273507. Time Dependence: A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
dt[n_] := With[{rules = RRules[n]}, Expand[Subtract[ Times[Expand[D[R[n] /. rules, Q]], Normal@Series[1/R[n], {k, 0, n}] /. rules, Cot[Q] ], 1]]]
dtCoefficients[n_] :=  With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
dtToEllK[NMax_] := ReplaceAll[-dt[NMax], {Cos[Q]^n_ :> Divide[Binomial[n, n/2], (2^(n))], k^n_ /; n > NMax -> 0} ]
Flatten[Numerator[dtCoefficients[10]]]
dtToEllK[5]

A274078 T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact differential time dependence.

Original entry on oeis.org

3, 15, 3, 315, 27, 27, 2835, 945, 27, 81, 155925, 2025, 2025, 135, 27, 6081075, 779625, 30375, 405, 243, 243, 638512875, 212837625, 654885, 42525, 8505, 1215, 729, 10854718875, 638512875, 58046625, 4465125, 127575, 3645, 729, 729

Offset: 1

Bradley Klee, Jun 09 2016

Triangle read by rows (see example). Comments of A274076 give a definition of the fraction triangle, which determines to arbitrary precision the differential time dependence for the time-independent solution (cf. A273506, A273507) of the plane pendulum's equations of motion. For more details see "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n\m|    1    2    3    4
---+---------------------
1  |    3;
2  |   15,   3;
3  |  315,  27,  27;
4  | 2835, 945,  27,  81;

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

Numerators: A274076. Phase Space Trajectory: A273506, A273507. Time Dependence: A274130, A274131. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
dt[n_] := With[{rules = RRules[n]}, Expand[Subtract[ Times[Expand[D[R[n] /. rules, Q]], Normal@Series[1/R[n], {k, 0, n}] /. rules, Cot[Q] ], 1]]]
dtCoefficients[n_] :=  With[{dtn = dt[n]}, Function[{a}, Coefficient[ Coefficient[dtn, k^a], Cos[Q]^(2 (a + #))] & /@ Range[a]] /@ Range[n]]
Flatten[Denominator[dtCoefficients[10]]]

A274130 Irregular triangle T(n,m), numerators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.

Original entry on oeis.org

1, 1, 11, 29, 1, 1, 491, 863, 6571, 4399, 13, 5, 1568551, 28783, 45187, 312643, 4351, 1117, 17, 35, 25935757, 81123251, 2226193, 2440117, 16025, 34246631, 18161, 35443, 49, 7, 5301974777, 22870237, 1603483793, 23507881213, 122574691, 122330761339, 903325919, 1976751869, 956873, 18551, 35, 77

Offset: 1

Bradley Klee, Jun 10 2016

Irregular triangle read by rows ( see examples ). The row length sequence is 2*n = A005843(n), n >= 1.The denominators are given in A274131.

The triangles A274076 and A274078 give the coefficients for the exact, differential time dependence of the plane pendulum's equations of motion. Integrating, we obtain time dependence as a Fourier sine series: t = -( (2/pi)K(k) Q + sum k^n * (T(n,m)/A274131(n,m)) * sin(2 m Q) ); where the sum runs over n = 1,2,3 ... and m = 1,2,3,...,2 n. Combining the phase space trajectory and time dependence, it is possible to express Jacobian elliptic functions {cn,dn} in parametric form. For more details read "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n\m  1     2     3      4    5   6 ...
-----------------------------------------
1  | 1    1
2  | 11   29    1      1
3  | 491  863   6571   4399  13  5
row n=4: 1568551, 28783, 45187, 312643, 4351, 1117, 17, 35,
row n=5: 25935757, 81123251, 2226193, 2440117, 16025, 34246631, 18161, 35443, 49, 7.
-----------------------------------------
The rational irregular triangle T(n, m) / A274131(n, m) begins:
n\m    1          2           3             4            5         6
-----------------------------------------------------------------------------
1  |  1/6,      1/48
2  |  11/96,    29/960,    1/160,          1/1536
3  |  491/5760, 863/30720, 6571/725760, 4399/1935360, 13/34560, 5/165888
row n=4: 1568551/23224320, 28783/1161216, 45187/4644864, 312643/92897280, 4351/4644864, 1117/5806080, 17/663552, 35/21233664,
row n=5: 25935757/464486400, 81123251/3715891200, 2226193/232243200, 2440117/619315200, 16025/11354112, 34246631/81749606400, 18161/185794560, 35443/2123366400, 49/26542080, 7/70778880.
-----------------------------------------------------------------------------
t1(Q) =-Q -(1/4)*k*Q -k*((1/6)*Sin[2*Q]+(1/48)*Sin[4*Q])+...
(2/Pi) K(k) ~ (1/(2 Pi)) t1(-2*Pi) =  1+(1/4)*k+...

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

Denominators: A274131. Phase Space Trajectory: A273506, A273507. Time Dependence: A274076, A274078. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
dt[n_] := With[{rules = RRules[n]}, Expand[Subtract[ Times[Expand[D[R[n] /. rules, Q]], Normal@Series[1/R[n], {k, 0, n}] /. rules, Cot[Q] ], 1]]]
t[n_] := Expand[ReplaceAll[Q TrigReduce[dt[n]], Cos[x_ Q] :> (1/x/Q) Sin[x Q]]]
tCoefficients[n_] := With[{tn = t[n]},Function[{a}, Coefficient[Coefficient[tn, k^a], Sin[2 # Q] ] & /@ Range[2 a]] /@ Range[n]]
tToEllK[NMax_]:= Expand[((t[NMax] /. Q -> -2 Pi)/2/Pi) /. k^n_ /; n > NMax -> 0]
Flatten[Numerator[-tCoefficients[10]]]
tToEllK[5]

A274131 Irregular triangle T(n,m), denominators of coefficients in a power/Fourier series expansion of the plane pendulum's exact time dependence.

Original entry on oeis.org

6, 48, 96, 960, 160, 1536, 5760, 30720, 725760, 1935360, 34560, 165888, 23224320, 1161216, 4644864, 92897280, 4644864, 5806080, 663552, 21233664, 464486400, 3715891200, 232243200, 619315200, 11354112, 81749606400, 185794560, 2123366400, 26542080, 70778880

Offset: 1

Bradley Klee, Jun 10 2016

Irregular triangle read by rows (see example). The row length sequence is 2*n = A005843(n), n >= 1.
  The numerator triangle is A274130.
Comments of A274130 give a definition of the fraction triangle, which determines to arbitrary precision the time dependence for the time-independent solution (cf. A273506, A273507) of the plane pendulum's equations of motion. For more details see "Plane Pendulum and Beyond by Phase Space Geometry" (Klee, 2016).

			n\m  1      2      3          4       5       6
------------------------------------------------------
1  | 6     48
2  | 96    960    160      1536
3  | 5760  30720  725760   1935360  34560   165888
------------------------------------------------------
row 4: 23224320, 1161216, 4644864, 92897280, 4644864, 5806080, 663552, 21233664,
row 5: 464486400, 3715891200, 232243200, 619315200, 11354112, 81749606400, 185794560, 2123366400, 26542080, 70778880.

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

Numerators: A274130. Phase Space Trajectory: A273506, A273507. Time Dependence: A274076, A274078. Elliptic K: A038534, A056982. Cf. A000984, A001790, A038533, A046161, A273496.

R[n_] := Sqrt[4 k] Plus[1, Total[k^# R[#, Q] & /@ Range[n]]]
Vq[n_] :=  Total[(-1)^(# - 1) (r Cos[Q] )^(2 #)/((2 #)!) & /@ Range[2, n]]
RRules[n_] :=  With[{H = ReplaceAll[1/2 r^2 + (Vq[n + 1]), {r -> R[n]}]},
Function[{rules}, Nest[Rule[#[[1]], ReplaceAll[#[[2]], rules]] & /@ # &, rules, n]][
   Flatten[R[#, Q] ->  Expand[(-1/4) ReplaceAll[ Coefficient[H, k^(# + 1)], {R[#, Q] -> 0}]] & /@ Range[n]]]]
dt[n_] := With[{rules = RRules[n]}, Expand[Subtract[ Times[Expand[D[R[n] /. rules, Q]], Normal@Series[1/R[n], {k, 0, n}] /. rules, Cot[Q] ], 1]]]
t[n_] := Expand[ReplaceAll[Q TrigReduce[dt[n]], Cos[x_ Q] :> (1/x/Q) Sin[x Q]]]
tCoefficients[n_] := With[{tn = t[n]},Function[{a}, Coefficient[Coefficient[tn, k^a], Sin[2 # Q] ] & /@ Range[2 a]] /@ Range[n]]
Flatten[Denominator[-tCoefficients[10]]]

A110258 Denominators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.

Original entry on oeis.org

1, 4, 64, 256, 16384, 65536, 1048576, 4194304, 1073741824, 4294967296, 68719476736, 274877906944, 17592186044416, 70368744177664, 1125899906842624, 4503599627370496, 4611686018427387904

Offset: 1

Paul D. Hanna, Jul 18 2005

Limit A110257(n)/a(n) = limit A110255(2*n-1)/A110256(2*n-1) = 4/Pi.

Apart from offset, identical to A056982.

			arctan(1/x) = 1/x - 1/(3*x^3) + 1/(5*x^5) - 1/(7*x^7) +-...
= [0; x, 3*x, 5/4*x, 28/9*x, 81/64*x, 704/225*x, 325/256*x,
768/245*x, 20825/16384*x, 311296/99225*x, 83349/65536*x,
1507328/480249*x, 1334025/1048576*x, 3145728/1002001*x,...]
= 1/(x + 1/(3*x + 1/(5/4*x + 1/(28/9*x + 1/(81/64*x +...))))).
The coefficients of x in the even-indexed partial quotients converge to Pi:
{3, 28/9, 704/225, 768/245, 311296/99225, ...}.
The coefficients of x in the odd-indexed partial quotients converge to 4/Pi:
{1, 5/4, 81/64, 325/256, 20825/16384, ...}.

Paul D. Hanna, Table of n, a(n) for n = 1..200
Index to divisibility sequences

See A056982 for another version of this sequence.

Cf. A110257 (numerators), A110255/A110256 (continued fraction), A110259/A110260.

{a(n)=denominator(subst((contfrac( sum(k=0,2*n+1,(-1)^k/x^(2*k+1)/(2*k+1)),2*n+2))[2*n],x,1))}

a(n)=4^(2*n-vecsum(binary(n-1))-2) \\ Charles R Greathouse IV, Apr 09 2012

a(n) = 4^A005187(n-1).

a(n) = A110256(2*n-1).

Edited by N. J. A. Sloane, Jun 05 2007

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