A001790 -id:A001790 - cp's OEIS frontend

A004731 a(0) = 1; thereafter a(n) = denominator of (n-2)!! / (n-1)!!.

1, 1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864, 300540195, 2147483648, 9917826435, 4294967296, 20419054425

Offset: 0

N. J. A. Sloane

Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
Also rational part of numerator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A036039).
Let x(m) = x(m-2) + 1/x(m-1) for m >= 3, with x(1)=x(2)=1. Then the numerator of x(n+2) equals the denominator of n!!/(n+1)!! for n >= 0, where the double factorials are given by A006882. - Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010, as corrected in Cooper (2015).
Numerator of (n-1)/( (n-2)/( .../1)), with an empty fraction taken to be 1. - Flávio V. Fernandes, Jan 31 2025

			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.

T. D. Noe, Table of n, a(n) for n=0..302
Joseph E. Cooper III, A recurrence for an expression involving double factorials, arXiv:1510.00399 [math.CO], 2015.
Svante Janson, On the traveling fly problem, Graph Theory Notes of New York Vol. XXXI, 17, 1996.

Cf. A001803, A004730, A006882 (double factorials), A036069.

Cf. A036039, A046161, A001790, A001803, A101926.

import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
   ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011

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));
#
[1, seq(denom(doublefactorial(n-2)/doublefactorial(n-1)), n = 1..36)]; # Peter Luschny, Feb 09 2025

Table[ Denominator[ (n-2)!! / (n-1)!! ], {n, 0, 31}] (* Jean-François Alcover, Jul 16 2012 *)
Denominator[#[[1]]/#[[2]]&/@Partition[Range[-2,40]!!,2,1]] (* Harvey P. Dale, Nov 27 2014 *)
Join[{1},Table[Numerator[(n/2-1/2)!/((n/2-1)!Sqrt[Pi])], {n,1,31}]] (* Peter Luschny, Feb 08 2025 *)

f(n) = prod(i=0, (n-1)\2, n - 2*i); \\ A006882
a(n) = if (n==0, 1, denominator(f(n-2)/f(n-1))); \\ Michel Marcus, Feb 08 2025

from sympy import gcd, factorial2
def A004731(n):
    if n <= 1:
        return 1
    a, b = factorial2(n-2), factorial2(n-1)
    return b//gcd(a,b) # Chai Wah Wu, Apr 03 2021

Name corrected by Michel Marcus, Feb 08 2025

A101926 a(n) = 2^A101925(n).

Original entry on oeis.org

2, 4, 16, 32, 256, 512, 2048, 4096, 65536, 131072, 524288, 1048576, 8388608, 16777216, 67108864, 134217728, 4294967296, 8589934592, 34359738368, 68719476736, 549755813888, 1099511627776, 4398046511104, 8796093022208

Offset: 0

Ralf Stephan, Dec 28 2004

nonn

a(n) is the numerator of 2^(2*n+1)*(n!)^2/(2*n+1)/(2*n)!. The corresponding denominator is A001803. - Daniel Suteu, Feb 03 2017
a(n) is the numerator of Integral_{x=-oo..oo} sech(x)^(2*n+2) dx. The corresponding denominator is A001803(n). - Mohammed Yaseen, Jul 25 2023
a(n) is the denominator of (1/Pi) * Integral_{x=0..Pi/2} sin(x)^(2*n) dx. The corresponding numerator is A001790(n). - Mohammed Yaseen, Sep 19 2023
a(n) = numerator(Pi*binomial(n, -1/2)). - Peter Luschny, Dec 05 2024

Index to divisibility sequences

Bisection of A036069 and of A086117.

Cf. A001803, A001790, A101925.

denom((binomial(2n,n)*4^-n)/2); # Stephen Crowley, Mar 05 2007

Table[Numerator[Beta[1, n + 1, 1/2]], {n, 0, 22}] (* Gerry Martens, Nov 13 2016 *)

More terms from Joshua Zucker, May 15 2006

A244419 Coefficient triangle of polynomials related to the Dirichlet kernel. Rising powers. Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2)).

Original entry on oeis.org

1, 1, 2, -1, 2, 4, -1, -4, 4, 8, 1, -4, -12, 8, 16, 1, 6, -12, -32, 16, 32, -1, 6, 24, -32, -80, 32, 64, -1, -8, 24, 80, -80, -192, 64, 128, 1, -8, -40, 80, 240, -192, -448, 128, 256, 1, 10, -40, -160, 240, 672, -448, -1024, 256, 512, -1, 10, 60, -160, -560, 672, 1792, -1024, -2304, 512, 1024

Offset: 0

Wolfdieter Lang, Jul 29 2014

This is the row reversed version of A180870. See also A157751 and A228565.
The Dirichlet kernel is D(n,x) = Sum_{k=-n..n} exp(i*k*x) = 1 + 2*Sum_{k=1..n} T(n,x) = S(n, 2*y) + S(n-1, 2*y) = S(2*n, sqrt(2*(1+y))) with y = cos(x), n >= 0, with the Chebyshev polynomials T (A053120) and S (A049310). This triangle T(n, k) gives in row n the coefficients of the polynomial Dir(n,y) = D(n,x=arccos(y)) = Sum_{m=0..n} T(n,m)*y^m. See A180870, especially the Peter Bala comments and formulas.
This is the Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2)) due to the o.g.f. for Dir(n,y) given by (1+z)/(1 - 2*y*z + z^2) = G(z)/(1 - y*F(z)) with G(z) = (1+z)/(1+z^2) and F(z) = 2*z/(1+z^2) (see the Peter Bala formula under A180870). For Riordan triangles and references see the W. Lang link 'Sheffer a- and z- sequences' under A006232.
The A- and Z- sequences of this Riordan triangle are (see the mentioned W. Lang link in the preceding comment also for the references): The A-sequence has o.g.f. 1+sqrt(1-x^2) and is given by A(2*k+1) = 0 and A(2*k) [2, -1/2, -1/8, -1/16, -5/128, -7/256, -21/1024, -33/2048, -429/32768, -715/65536, ...], k >= 0. (See A098597 and A046161.)
  The Z-sequence has o.g.f. sqrt((1-x)/(1+x)) and is given by
  [1, -1, 1/2, -1/2, 3/8, -3/8, 5/16, -5/16, 35/128, -35/128, ...]. (See A001790 and A046161.)
The column sequences are A057077, 2*(A004526 with even numbers signed), 4*A008805 (signed), 8*A058187 (signed), 16*A189976 (signed), 32*A189980 (signed) for m = 0, 1, ..., 5.
The row sums give A005408 (from the o.g.f. due to the Riordan property), and the alternating row sums give A033999.
The row polynomials Dir(n, x), n >= 0, give solutions to the diophantine equation (a + 1)*X^2 - (a - 1)*Y^2 = 2 by virtue of the identity (a + 1)*Dir(n, -a)^2 - (a - 1)*Dir(n, a)^2 = 2, which is easily proved inductively using the recurrence Dir(n, a) = (1 + a)*(-1)^(n-1)*Dir(n-1, -a) + a*Dir(n-1, a) given below by Wolfdieter Lang. - Peter Bala, May 08 2025

			The triangle T(n,m) begins:
  n\m  0   1   2    3    4    5    6     7     8    9    10 ...
  0:   1
  1:   1   2
  2:  -1   2   4
  3:  -1  -4   4    8
  4:   1  -4 -12    8   16
  5:   1   6 -12  -32   16   32
  6:  -1   6  24  -32  -80   32   64
  7:  -1  -8  24   80  -80 -192   64   128
  8:   1  -8 -40   80  240 -192 -448   128   256
  9:   1  10 -40 -160  240  672 -448 -1024   256  512
  10: -1  10  60 -160 -560  672 1792 -1024 -2304  512  1024
  ...
Example for A-sequence recurrence: T(3,1) = Sum_{j=0..2} A(j)*T(2,j) = 2*(-1) + 0*2 + (-1/2)*4 = -4. Example for Z-sequence recurrence: T(4,0) = Sum_{j=0..3} Z(j)*T(3,j) = 1*(-1) + (-1)*(-4) + (1/2)*4 + (-1/2)*8 = +1. (For the A- and Z-sequences see a comment above.)
Example for the alternate recurrence: T(4,2) = 2*T(3,1) - T(3,2) = 2*(-4) - 4 = -12. T(4,3) = 0*T(3,2) + T(3,3) = T(3,3) = 8. - _Wolfdieter Lang_, Jul 30 2014

Wikipedia, Dirichlet kernel.

Dir(n, x) : A005408 (x = 1), A002878 (x = 3/2), A001834 (x = 2), A030221 (x = 5/2), A002315 (x = 3), A033890 (x = 7/2), A057080 (x = 4), A057081 (x = 9/2), A054320 (x = 5), A077416 (x = 6), A028230 (x = 7), A159678 (x = 8), A049629 (x = 9), A083043 (x = 10),
(-1)^n * Dir(n, x): A122367 (x = -3/2); A079935 (x = -2), A004253 (x = -5/2), A001653 (x = -3), A049685 (x = -7/2), A070997 (x = -4), A070998 (x = -9/2), A072256(n+1) (x = -5).
Cf. A180870 (row reversed), A157751, A228565, A049310, A053120, A057077, A004526, A008805, A058187, A189976, A189980, A005408, A033999.

T[n_, k_] := T[n, k] = Which[k == 0, (-1)^Quotient[n, 2], (0 <= n && n < k) || (n == -1 && k == 1), 0, True, 2 T[n-1, k-1] - T[n-2, k]];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2019, from Sage *)

def T(n, k):
    if k == 0: return (-1)^(n//2)
    if (0 <= n and n < k) or (n == -1 and k == 1): return 0
    return 2*T(n-1, k-1) - T(n-2, k)
for n in range(11): [T(n,k) for k in (0..n)] # Peter Luschny, Jul 29 2014

T(n, m) = [y^m] Dir(n,y) for n >= m >= 0 and 0 otherwise, with the polynomials Dir(y) defined in a comment above.
T(n, m) = 2^m*(S(n,m) + S(n-1,m)) with the entries S(n,m) of A049310 given there explicitly.
O.g.f. for polynomials Dir(y) see a comment above (Riordan triangle ((1+z)/(1+z^2), 2*z/(1+z^2))).
O.g.f. for column m: ((1 + x)/(1 + x^2))*(2*x/(1 + x^2))^m, m >= 0, (Riordan property).
Recurrence for the polynomials: Dir(n, y) = 2*y*Dir(n-1, y) - Dir(n-2, y), n >= 1, with input D(-1, y) = -1 and D(0, y) = 1.
Triangle three-term recurrence: T(n,m) = 2*T(n-1,m-1) - T(n-2,m) for n >= m >= 1 with T(n,m) = 0 if 0 <= n < m, T(0,0) = 1, T(-1,1) = 0 and T(n,0) = A057077(n) = (-1)^(floor(n/2)).
From Wolfdieter Lang, Jul 30 2014: (Start)
In analogy to A157751 one can derive a recurrence for the row polynomials Dir(n, y) = Sum_{m=0..n} T(n,m)*y^m also using a negative argument but only one recursive step: Dir(n,y) = (1+y)*(-1)^(n-1)*Dir(n-1,-y) + y*Dir(n-1,y), n >= 1, Dir(0,y) = 1 (Dir(-1,y) = -1). See also A180870 from where this formula can be derived by row reversion.
This entails another triangle recurrence T(n,m) = (1 + (-1)^(n-m))*T(n-1,m-1) - (-1)^(n-m)*T(n-1,m), for n >= m >= 1 with T(n,m) = 0 if n < m and T(n,0) = (-1)^floor(n/2). (End)
From Peter Bala, Aug 14 2022: (Start)
The row polynomials Dir(n,x), n >= 0, are related to the Chebyshev polynomials of the first kind T(n,x) by the binomial transform as follows:
(2^n)*(x - 1)^(n+1)*Dir(n,x) = (-1) * Sum_{k = 0..2*n+1} binomial(2*n+1,k)*T(k,-x).
Note that Sum_{k = 0..2*n} binomial(2*n,k)*T(k,x) = (2^n)*(1 + x)^n*T(n,x). (End)
From Peter Bala, May 04 2025: (Start)
For n >= 1, the n-th row polynomial Dir(n, x) = (-1)^n * (U(n, -x) - U(n-1, -x)) = U(2*n, sqrt((1+x)/2)), where U(n, x) denotes the n-th Chebyshev polynomial of the second kind.
For n >= 1 and x < 1, Dir(n, x) = (-1)^n * sqrt(2/(1 - x )) * T(2*n+1, sqrt((1 - x)/2)), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind.
Dir(n, x)^2 - 2*x*Dir(n, x)*Dir(n+1, x) + Dir(n+1, x)^2 = 2*(1 + x).
Dir(n, x) = (-1)^n * R(n, -2*(x+1)), where R(n, x) is the n-th row polynomial of the triangle A085478.
Dir(n, x) = Sum_{k = 0..n} (-1)^(n+k) * binomial(n+k, 2*k) * (2*x + 2)^k. (End)

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]]]

A161199 Numerators in expansion of (1-x)^(-5/2).

Original entry on oeis.org

1, 5, 35, 105, 1155, 3003, 15015, 36465, 692835, 1616615, 7436429, 16900975, 152108775, 339319575, 1502700975, 3305942145, 115707975075, 251835004575, 1091285019825, 2354878200675, 20251952525805, 43397041126725, 185423721177825, 395033145117975

Offset: 0

Johannes W. Meijer, Jun 08 2009

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A161198 (triangle for (1-x)^((-1-2*n)/2) for all values of n).
Cf. A046161 (denominators for (1-x)^(-5/2)).
Numerators of [x^n]( (1-x)^(p/2) ): A161202 (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), this sequence (p=-5), A161201 (p=-7).

A161199:= func< n | Numerator( Binomial(n+3,3)*Catalan(n+2)/2^(2*n+1) ) >;
[A161199(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(-5/2),{x,0,30}],x]] (* or *) Numerator[Table[(4n^2+8n+3)/3 Binomial[2n,n]/4^n,{n,0,30}]] (* Harvey P. Dale, Oct 15 2011 *)

def A161199(n): return numerator((-1)^n*binomial(-5/2,n))
[A161199(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator(((3 + 8*n + 4*n^2)/3)*binomial(2*n,n)/(4^n)).

a(n) = denominator((3/2)*Integral_{x=0..1} x^n*sqrt(1-x) dx), where the integral is sqrt(Pi)*n!/Gamma(n+5/2) = n!/( (n+3/2)*(n+1/2)*(n-1/2)*...*(1/2)). - Groux Roland, Feb 23 2011

A161202 Numerators in expansion of (1-x)^(5/2).

Original entry on oeis.org

1, -5, 15, -5, -5, -3, -5, -5, -45, -55, -143, -195, -1105, -1615, -4845, -7429, -185725, -294975, -950475, -1550775, -10235115, -17058525, -57378675, -97294275, -1329688425, -2287064091, -7916760315, -13781027215

Offset: 0

Johannes W. Meijer, Jun 08 2009

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

Cf. A046161 (denominators).
Cf. A161198 (triangle of coefficients of (1-x)^((-1-2*n)/2)).
Numerators of [x^n]( (1-x)^(p/2) ): this sequence (p=5), A161200 (p=3), A002596 (p=1), A001790 (p=-1), A001803 (p=-3), A161199 (p=-5), A161201 (p=-7).

A161202:= func< n | -Numerator(15*(n+1)*Catalan(n)/(4^n*(2*n-1)*(2*n-3)*(2*n-5))) >;
[A161202(n): n in [0..30]]; // G. C. Greubel, Sep 24 2024

Numerator[CoefficientList[Series[(1-x)^(5/2),{x,0,30}],x]] (* Harvey P. Dale, Aug 22 2011 *)
Table[(-1)^n*Numerator[Binomial[5/2, n]], {n,0,30}] (* G. C. Greubel, Sep 24 2024 *)

def A161202(n): return (-1)^n*numerator(binomial(5/2,n))
[A161202(n) for n in range(31)] # G. C. Greubel, Sep 24 2024

a(n) = numerator( (15/(15-46*n+36*n^2-8*n^3))*binomial(2*n,n)/(4^n) ).

a(n) = (-1)^n*numerator( binomial(5/2, n) ). - G. C. Greubel, Sep 24 2024

cp's OEIS Frontend

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