A159601 -id:A159601 - cp's OEIS frontend

A159605 E.g.f: Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)! = Series_Reversion of e.g.f. S(x) of A159601.

1, 3, 63, 3465, 363825, 62214075, 15740160975, 5524796502225, 2569030373534625, 1528573072253101875, 1132672646539548489375, 1022803399825212285905625, 1105650475211054481063980625, 1409704355894094463356575296875

Offset: 1

Paul D. Hanna, May 11 2009

nonn

			E.g.f.: A(x) = x + 3*x^3/3! + 63*x^5/5! + 3465*x^7/7! +...
A(S(x)) = x where S(x) = Sum_{n>=1} A159601(n)*x^(2n-1)/(2n-1)! :
S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...

Harvey P. Dale, Table of n, a(n) for n = 1..213

Cf. A159601.

Table[Product[(2k-3)(4k-5),{k,n}],{n,15}] (* Harvey P. Dale, Jan 31 2023 *)

PARI
```
a(n)=prod(k=1,n,(2*k-3)*(4*k-5))
```

a(n) = Product_{k=1..n} (2k-3)(4k-5).

a(n) ~ Gamma(1/4) * 2^(3*n - 5/2) * n^(2*n - 7/4) / (sqrt(Pi) * exp(2*n)). - Vaclav Kotesovec, Nov 19 2023

A159600 E.g.f. C(x) satisfies: C(x) = (1 - 2*S(x)^2)^(1/4), where S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

Original entry on oeis.org

1, -1, 3, -27, 441, -11529, 442827, -23444883, 1636819569, -145703137041, 16106380394643, -2164638920874507, 347592265948756521, -65724760945840254489, 14454276753061349098587

Offset: 0

Paul D. Hanna, May 07 2009

sign

See A104203 for the expansion of the sine lemniscate function sl(x).
E.g.f. C(x) is an even function; zero terms are omitted.
Radius of convergence is |x| <= r:
r = sqrt(2)*(Pi/2)^(3/2)/gamma(3/4)^2 with
C(r) = gamma(3/4)^2/(Pi/2)^(3/2) where:
r = L/sqrt(2) where L=Lemniscate constant;
r = 1.8540746773013719184338503471952600...
C(r) = 0.76275976350181318806232598096361579...

			E.g.f.: C(x) = 1 - x^2/2! + 3*x^4/4! - 27*x^6/6! + 441*x^8/8! -+ ...
C(x)^2 = 1 - 2*x^2/2! + 12*x^4/4! - 144*x^6/6! + 3024*x^8/8! -+ ...
C(x)^3 = 1 - 3*x^2/2! + 27*x^4/4! - 441*x^6/6! + 11529*x^8/8! -+ ...
C(x)^4 = 1 - 4*x^2/2! + 48*x^4/4! - 1008*x^6/6! + 32256*x^8/8! -+ ...
C(x)^4 + 2*S(x)^2 = 1 where:
S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! + ...
S(x)^2 = 2*x^2/2! - 24*x^4/4! + 504*x^6/6! - 16128*x^8/8! +-...
From _Paul D. Hanna_, Jul 29 2011: (Start)
O.g.f.: 1 - x + 3*x^2 - 27*x^3 + 441*x^4 - 11529*x^5 + 442827*x^6 -+ ... + a(n)*x^n + ...
O.g.f.: 1/(1 + x/(1 + 2*x/(1 + 9*x/(1 + 8*x/(1 + 25*x/(1 + 18*x/(1 + 49*x/(1 + 32*x/(1-...))))))))) (continued fraction). (End)

Cf. A159601 (S(x)); A193541, A193544: All of these have the same |a(n)|. - M. F. Hasler, Aug 31 2012

Cf. A129194.

a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, m! SeriesCoefficient[ JacobiCN[ x, 1/2], {x, 0, m}]]]; (* Michael Somos, Apr 25 2011 *)
a[ n_] := If[ n < 0, 0, With[ {m = 2 n}, With[ {s = InverseSeries[ Integrate[ Series[(1 - x^4 / 4) ^ (-1/2), {x, 0, m + 1}], x]]}, m! SeriesCoefficient[ Sqrt[ (2 - s^2) / (2 + s^2)], {x, 0, m}]]]]; (* Michael Somos, Apr 25 2011 *)

{a(n)=local(S=x,C);for(i=0,2*n,S=intformal((1-2*S^2+O(x^(2*n+2)))^(3/4))); C=(1-2*S^2)^(1/4) ;(2*n)!*polcoeff(C,2*n)}

{a(n) = my(A, m); if( n<0, 0, m = 2*n; A = serreverse( intformal( (1 - x^4 / 4 + x * O(x^m)) ^ (-1/2))); m! * polcoeff( sqrt( (2 - A^2) / (2 + A^2)), m))}; /* Michael Somos, Apr 25 2011 */

{a(n) = local(C=1+x); for(i=1,n, C = exp( intformal( C * intformal(-1/C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C,n)}
for(n=0,20,print1(a(2*n),", "))

{a(n) = local(C=1+x); for(i=1,n, C = exp( intformal( -1/C * intformal(C^3 + x*O(x^n)) ) ) ); n!*polcoeff(C,n)}
for(n=0,20,print1(a(2*n),", "))

E.g.f. C(x) satisfies: C(x)^4 + 2*S(x)^2 = 1 where S(x) = Integral [1 - 2*S(x)^2]^(3/4) dx with S(0)=0; Left-shift of the Laplace transform of e.g.f. C(x) equals the Laplace transform of S(x).
E.g.f.: Sum_{k>=0} 2^k * a(k) * x^(2*k) / (2*k)! = cos lemn(x) where cos lemn(x) is the cosine lemniscate function of Gauss. - Michael Somos, Apr 25 2011
O.g.f.: 1/(1 + 1^2*x/(1 + 2^2/2*x/(1 + 3^2*x/(1 + 4^2/2*x/(1 + 5^2*x/(1 + 6^2/2*x/(1 + 7^2*x/(1 + 8^2/2*x/(1+...))))))))) (continued fraction). - Paul D. Hanna, Jul 29 2011
a(n) = (-1)^floor(n/2) * A193544(n) = (-1)^ceiling(n/2) * A193544(n) = -A159601(n). - M. F. Hasler, Aug 31 2012
G.f.: 1/Q(0) where p=1/2, Q(k) = 1 + x*(2*k+1)^2/( 1 + p*x*(2*k+2)^2/Q(k+1) ); (continued fraction due to Stieltjes T.J.). - Sergei N. Gladkovskii, Mar 22 2013
From Paul D. Hanna, Jun 02 2015: (Start)
E.g.f. C(x) satisfies:
(1) C(x) = exp( Integral C(x) * Integral -1/C(x)^3 dx dx ).
(2) C(x) = exp( Integral -1/C(x) * Integral C(x)^3 dx dx ).
(End)
G.f.: 1 / (1 + b(1)*x / (1 + b(2)*x / (1 + b(3)*x / ... ))) where b = A129194. - Michael Somos, Jan 03 2013

A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

Original entry on oeis.org

1, 1, 1, 0, -3, -12, -27, 0, 441, 3024, 11529, 0, -442827, -4390848, -23444883, 0, 1636819569, 21224560896, 145703137041, 0, -16106380394643, -257991277243392, -2164638920874507, 0, 347592265948756521

Offset: 0

Peter Luschny, Jul 26 2011

sign

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972.
Eric W. Weisstein, Jacobi Elliptic Functions.

Cf. A060627, A060628, A104203, A159600, A193541, A193544.

A190904 := proc(n) option remember; `if`(n=0,1,add(((1-irem(k,2))*(-1)^ iquo(k,2))*binomial(n-1,k)*A190904(n-1-k)*A190904(k),k=0..n-1)) end:

a[0] = 1;
a[n_] := a[n] =
  Sum[Mod[(k+1)^3, 4, -1] Binomial[n-1, k] a[n-k-1] a[k], {k, 0, n-1}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 24 2019 *)

Let F(n,x) = Sum_{k=0..n-1} cos(Pi*k*x)*binomial(n-1,k)*F(n-1-k,x)* F(k,x), then
F(n, 0)   = n! = A000142(n),
F(n, 1/2) = a(n),
F(n, 1)   = 2^n*Euler_{n}(1) = A_{n}(-1) = A155585(n).
a(2*n) = A159601(n)*(-1)^floor((n-1)/2).
a(2*n+1) = A104203(2*n+1).
From Peter Bala, Aug 25 2011: (Start)
The sequence entries may be calculated as follows: Define the nested derivative D^n[f](x) by means of the recursion D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. The coefficients in the expansion of D^n[f](x) in powers of f(x) can be found in A145271. Then we have
a(2*n) = D^(2*n)[sqrt(1+sin^2(x))](0)
a(2*n+1) = D^(2*n)[sqrt(1-x^4)](0).
The generating function involves the Jacobian elliptic functions. Define E(u,k) := cn(i*u,k)-i*sn(i*u,k) = 1+u+u^2/2!+(1+k^2)*u^3/3!+(1+4*k^2)*u^4/4!+..., where cn(u,k) and sn(u,k) are Jacobian elliptic functions of modulus k (see A060627 and A060628). Then the e.g.f. A(u) for this sequence is
A(u) := E(u,i) = 1+u+u^2/2!-3*u^4/4!-12*u^5/5!-27*u^6/6!+....
Proof: Using well-known properties of the Jacobian elliptic functions (see for example Abramowitz and Stegun, Chapter 16) we find the generating function A(u) satisfies the differential equation
(d/du)A(u) = dn(i*u,i)*A(u) = 1/2*(A(i*u)+A(-i*u))*A(u), which leads to a recurrence for the coefficients of A(u):
a(n+1) = sum{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*a(2*k)*a(n-2*k) with a(0) = 1. This recurrence is equivalent to the defining recurrence for this sequence given above.
End proof.
The generating function A(u) satisfies 1/A(u) = A(-u).
Compare entries of this sequence with those of A104203, A159600, A193541 and A193544.
(End)

A286306 a(n) = coefficient of x^(2n)/(2n)! in exp( integral ( sn(x, 1/2) / cd(x, 1/2) ) dx).

Original entry on oeis.org

1, 1, 3, 27, 441, 11529, 442827, 23444883, 1636819569, 145703137041, 16106380394643, 2164638920874507, 347592265948756521, 65724760945840254489, 14454276753061349098587, 3658147171522531111996803, 1055646229815910768764248289, 344553616791279239828059918881

Offset: 0

Michael Somos, May 05 2017

nonn

			G.f. = 1 + x + 3*x^2 + 27*x^3 + 441*x^4 + 11529*x^5 + 442827*x^6 + ...
E.g.f. = 1 + 1*x^2/2! + 3*x^4/4! + 27*x^6/6! + 441*x^8/8! + 11529*x^10/10! + ...

G. C. Greubel, Table of n, a(n) for n = 0..249

Cf. A159600, A159601, A190904, A193541, A193544, A242240.

a[ n_] := If[ n < 0, 0, With[{m = 2 n}, m! SeriesCoefficient[ Exp[ Integrate[ JacobiSN[x, 1/2] / JacobiCD[x, 1/2], x]], {x, 0, m}]]];
a:= With[{nmax = 110}, CoefficientList[Series[Exp[Integrate[JacobiSN[x, 1/2]/JacobiCD[x, 1/2], x]], {x, 0, nmax}], x]*Range[0, nmax]!][[1 ;; ;; 2]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 29 2018 *)

{a(n) = my(m); if( n<0, 0, m = 2*n; m! * polcoeff( exp( intformal( serreverse( intformal( (1 + x^4 + x * O(x^m))^(-1/2))))), m))};

Given e.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n) / (2*n)!, then 0 = 1 + 2*A'^2 - A*A''.
Given e.g.f. A(x), then A'(x) / A(x) = B(x) where B() is the e.g.f. for A242240.
Given e.g.f. A(x), 1 / A(x) = A(-x).
A159600(n) = (-1)^n * a(n). A159601(n) = -(-1)^n * a(n) if n>0.
A190904(2*n) = A193541(n) = (-1)^floor(n/2) * a(n). A193544(n) = (-1)^floor((n+1)/2) * a(n).

cp's OEIS Frontend

A159605 E.g.f: Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)! = Series_Reversion of e.g.f. S(x) of A159601.

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A159600 E.g.f. C(x) satisfies: C(x) = (1 - 2*S(x)^2)^(1/4), where S'(x) = C(x)^3 and C'(x) = -S(x) with C(0)=1.

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A190904 a(n) = Sum_{k=0..n-1} cos(Pi*k/2)*binomial(n-1,k)*a(n-1-k)*a(k) for n > 0, a(0) = 1.

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A286306 a(n) = coefficient of x^(2*n)/(2*n)! in exp( integral ( sn(x, 1/2) / cd(x, 1/2) ) dx).

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A190904 a(n) = Sum_{k=0..n-1} cos(Pik/2)binomial(n-1,k)a(n-1-k)a(k) for n > 0, a(0) = 1.

A286306 a(n) = coefficient of x^(2n)/(2n)! in exp( integral ( sn(x, 1/2) / cd(x, 1/2) ) dx).