A012393 -id:A012393 - cp's OEIS frontend

A000831 Expansion of e.g.f. (1 + tan(x))/(1 - tan(x)).

1, 2, 4, 16, 80, 512, 3904, 34816, 354560, 4063232, 51733504, 724566016, 11070525440, 183240753152, 3266330312704, 62382319599616, 1270842139934720, 27507470234550272, 630424777638805504, 15250953398036463616, 388362339077351014400, 10384044045105304174592

Offset: 0

N. J. A. Sloane

nonn

			(1+tan x)/(1-tan x) = 1 + 2*x/1! + 4*x^2/2! + 16*x^3/3! + 80*x^4/4! + 512*x^5/5! + ...

Alois P. Heinz, Table of n, a(n) for n = 0..432 (first 84 terms from R. J. Mathar)
William Y. C. Chen and Amy M. Fu, The Dumont Ansatz for the Eulerian Polynomials, Peak Polynomials and Derivative Polynomials, arXiv:2204.01497 [math.CO], 2022.
D. Dumont, Further triangles of Seidel-Arnold type and continued fractions related to Euler and Springer numbers, Adv. Appl. Math., 16 (1995), 275-296.
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

Bisections: A002436 and A012393.

Cf. A000111, A000182, A155100, A258880, A258901, A258994.

m:=30; R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( (1+Tan(x))/(1-Tan(x)) ))); // G. C. Greubel, Mar 21 2019; Apr 28 2023

A000831 := (1+tan(x))/(1-tan(x)) : for n from 0 to 200 do printf("%d %d ",n,n!*coeftayl(A000831,x=0,n)) ; end: # R. J. Mathar, Nov 19 2006
A000831 := n -> `if`(n=0,1,(-1)^((n^2-n)/2)*4^n*(euler(n,1/2)+euler( n,1))): # Peter Luschny, Nov 24 2010
# third Maple program:
b:= proc(u, o) option remember;
      `if`(u+o=0, 1, 2*add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2020

Range[0, 18]! CoefficientList[Series[(1+Tan[x])/(1-Tan[x]), {x,0,18}], x] (* Robert G. Wilson v, Apr 16 2011 *)
b[u_, o_] := b[u, o] = If[u+o == 0, 1, 2*Sum[b[o-1+j, u-j], {j, 1, u}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Dec 02 2023, after Alois P. Heinz *)

a(n):=sum(if evenp(n+k) then ((-1)^((n+k)/2)*sum(j!*stirling2(n,j)*2^(n-j+1)*(-1)^(j)*binomial(j-1,k-1),j,k,n)) else 0,k,1,n); /* Vladimir Kruchinin, Aug 19 2010 */

a(n) = if( n<1, n==0, n! * polcoeff( 1 + 2 / ( 1 / tan( x + x * O(x^n)) - 1), n)) /* Michael Somos, Apr 16 2011 */

a(n) = local(A); if( n<0, 0, A = x * O(x^n); n! * polcoeff( (cos(x + A) + sin(x + A)) / (cos(x + A) - sin(x + A)), n)) /* Michael Somos, Apr 16 2011 */

@CachedFunction
def sp(n,x) :
    if n == 0 : return 1
    return -add(2^(n-k)*sp(k,1/2)*binomial(n,k) for k in range(n)[::2])
A000831 = lambda n : abs(sp(n,x))
[A000831(n) for n in (0..21)]     # Peter Luschny, Jul 30 2012

def A000831_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( (1+tan(x))/(1-tan(x)) ).egf_to_ogf().list()
A000831_list(40) # G. C. Greubel, Mar 21 2019; Apr 28 2023

E.g.f.: tan(x+Pi/4).
a(n) = Sum_{k=1..n} (if even(n+k) ( (-1)^((n+k)/2)*Sum_{j=k..n} (j!*stirling2(n,j)*2^(n-j+1)*(-1)^(j)*binomial(j-1,k-1) ), n>0. - Vladimir Kruchinin, Aug 19 2010
a(n) = 4^n*(E_{n}(1/2) + E_{n}(1))*(-1)^((n^2-n)/2) for n > 0, where E_{n}(x) is an Euler polynomial. - Peter Luschny, Nov 24 2010
a(n) = 2^n * A000111(n). - Gerard P. Michon, Feb 24 2011
From Sergei N. Gladkovskii, Dec 01 2011 - Jan 24 2014: (Start)
Continued fractions:
E.g.f.: -1 + 2/(1-x*G(0)); G(k) = 1 - (x^2)/((x^2) - (2*k + 1)*(2*k + 3)/G(k+1)).
E.g.f.: 1 + 2*x/(U(0)-2*x) where U(k) = 4*k+1 + x/(1+x/ (4*k+3 - x/(1- x/U(k+1)))).
E.g.f.: 1 + 2*x/(G(0)-x) where G(k) =  2*k+1 - x^2/G(k+1).
G.f.: 1 + 2*x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - 2*x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - 2*x^2*(2*k+2)*(2*k+3)/Q(k+1)).
E.g.f.: tan(2*x) + sec(2*x) = (x-1)/(x+1) - 2*(2*x^2+3)/( T(0)*3*x*(1+x)- 2*x^2-3)/(x+1), where T(k) = 1 - x^4*(4*k-1)*(4*k+7)/( x^4*(4*k-1)*(4*k+7) - (4*k+1)*(4*k+5)*(16*k^2 + 8*k - 2*x^2 - 3)*(16*k^2 + 40*k - 2*x^2 + 21)/T(k+1)).
E.g.f.: 1 + 2*x/Q(0), where Q(k) = 4*k+1 -x/(1 - x/( 4*k+3 + x/(1 + x/Q(k+1)))).
E.g.f.: tan(2*x) + sec(2*x) = 2*R(0)-1, where R(k) = 1 + x/( 4*k+1 - x/(1 - x/( 4*k+3 + x/R(k+1)))).
G.f.: 1+ G(0)*2*x/(1-2*x), where G(k) = 1 - 2*x^2*(k+1)*(k+2)/(2*x^2*(k+1)*(k+2) - (1-2*x*(k+1))*(1-2*x*(k+2))/G(k+1)). (End)
a(n) ~ n! * (4/Pi)^(n+1). - Vaclav Kotesovec, Jun 15 2015
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n-2,k) * a(k) * a(n-k-1). - Ilya Gutkovskiy, Jun 11 2020

A000825 Expansion of cos x (1 + sin x ) /cos 2x.

Original entry on oeis.org

1, 1, 3, 8, 57, 256, 2763, 17408, 250737, 2031616, 36581523, 362283008, 7828053417, 91620376576, 2309644635483, 31191159799808, 898621108880097, 13753735117275136, 445777636063460643, 7625476699018231808

Offset: 0

N. J. A. Sloane

nonn

R. J. Mathar, Table of n, a(n) for n = 0..200

Cf. A000828, A000831.

Bisections are A000281 and (1/2) * A012393.

With[{nn=20},CoefficientList[Series[Cos[x] (1+Sin[x])/Cos[2x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 08 2013 *)

a(n) ~ n! * (sqrt(2) + 1 + (sqrt(2)-1)*(-1)^n) * 4^n / Pi^(n+1). - Vaclav Kotesovec, Jun 01 2015

A092635 Coefficients in asymptotic expansion of (Pin/16^n)binomial(2n,n)^2 in powers of 1/(8n).

Original entry on oeis.org

1, -2, 2, 4, -10, -92, 212, 4744, -10330, -440492, 926972, 64905784, -133989700, -13958115992, 28507072232, 4129849252624, -8382195909370, -1609609189734092, 3254514296768492, 799400112196210264, -1612314122867559340

Offset: 0

Michael Somos, Mar 02 2004

sign

			binomial(2*n,n)^2*Pi*n/16^n ~ 1 -2/(8n) +2/(8n)^2 +4/(8n)^3 + ...
From _Paul D. Hanna_, Apr 02 2011: (Start)
O.g.f.: A(x) = 1 - 2*x + 2*x^2 + 4*x^3 - 10*x^4 - 92*x^5 + 212*x^6 + ...
E.g.f.: E(x) = 1 - 2*x + 2*x^2/2! + 4*x^3/3! - 10*x^4/4! - 92*x^5/5! + 212*x^6/6! + ...
Note that E(x)*exp(2*x) = sqrt(E(x)*E(-x)) is an even function:
E(x)*exp(2*x) = 1 - 2*x^2/2! + 22*x^4/4! - 692*x^6/6! + 45862*x^8/8! + ... (End)

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A002894, A012393, A188514.

#Uses the conjectural recurrence equation
A092635 := proc (n) option remember; if n = 0 then 1 else add((-4)^(n-k)*euler(n-k, 1)*A092635(k), k = 0 .. n-1)/n end if; end proc:
seq(A092635(n), n = 0 .. 20); # Peter Bala, Mar 12 2015

m = 21;
1/(1+2x/(1-x+ContinuedFractionK[(4i^2-1)x^2, 1, {i, 1, m/2 // Floor}]))+O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 02 2019 *)
CoefficientList[Normal[Series[Pi*n*Binomial[n/4, n/8]^2 / 2^(n/2+3), {n, Infinity, 20}]], 1/n] (* Vaclav Kotesovec, Aug 18 2024 *)

a(n)=if(n<0,0,polcoeff(exp(intformal(serlaplace(-1/cosh(x*2+x*O(x^n))^2*2))),n))

a(n)=local(A);if(n<1,n==0,A=1; for(m=1,n,A+=x^2*O(x^m);A=Pol(A)+polcoeff(subst(A,x,x/(1+8*x))/A*(1+8*x)/(1+4*x)^2,m+1)/8/m*x^m);polcoeff(A,n))

{B(x,n) = prod(k=1,n, (1 + 4*k*x)^((-1)^k*binomial(n,k)) )}
{a(n) = polcoeff( A = prod(m=1,n, B(x  +x*O(x^n), m)^(1/2^m)), n)}
for(n=0,30,print1(a(n),", ")) \\ Paul D. Hanna, Mar 03 2024

E.g.f. E(x) satisfies: E(x) = E(-x)*exp(-4*x). - Paul D. Hanna, Apr 02 2011
O.g.f.: A(x) = 1/(1 + 2*x/(1-x + 3*x^2/(1 + 15*x^2/(1 + 35*x^2/(1 + 63*x^2/(1 + 99*x^2/(1 + ...+ (4*n^2-1)*x^2/(1 + ...)))))))) (continued fraction). - Paul D. Hanna, Apr 02 2011
From Peter Bala, Mar 12 2015: (Start)
Let 2/(1 + exp(4*x)) = Sum_{n >= 0} b(n)*x^n/n!, so that b(n) = (-4)^n*E(n,1), where E(n,x) is the n-th Euler polynomial.
Then exp( Sum_{n >= 1} b(n)*x^n/n ) = 1 - 2*x + 2*x^2 + 4*x^3 - 10*x^4 - 92*x^5 + ... appears to be the o.g.f. for this sequence.
Assuming this is true we have the recurrence a(0) = 1 and a(n) = 1/n*Sum_{k = 0..n-1} (-4)^(n-k)*E(n-k,1)*a(k) for n >= 1. (End)
O.g.f.: A(x) = Product_{n>=1} B(x,n)^(1/2^n) where B(x,n) = Product{k=1..n} (1 + 4*k*x)^((-1)^k*binomial(n,k)). Compare to Product_{n>=1} B(x,n) = 1 - 4*x and Product_{n>=1} B(x,n)^(1/n) = exp(-4*x). - Paul D. Hanna, Mar 03 2024
Conjecture from Paul D. Hanna, Aug 15 2024: (Start)
Let A(x) = Sum_{n>=0} a(n)*x^(n+1), then A(x) satisfies:
  A(x) = -x^2 / A(-x),
  A(x) = -A(-x/(1+4*x)),
  A(x) = x^2 / A(x/(1-4*x)). (End)

A253165 a(n) = (-1)^n2^(6n+3)(zeta(-2n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.

Original entry on oeis.org

1, 8, 256, 17408, 2031616, 362283008, 91620376576, 31191159799808, 13753735117275136, 7625476699018231808, 5192022022552652087296, 4258996468871236847403008, 4142655008190840426050093056, 4714505177821257067736657297408, 6206008749802659037752564348092416

Offset: 0

Peter Luschny, Mar 11 2015

nonn

Cf. A000825, A000828, A000831, A012393.

a := n -> (-1)^n*2^(6*n+3)*(Zeta(0,-2*n-1,1/2)-Zeta(0,-2*n-1, 1)):
seq(a(n), n=0..14);

f[n_] := (-1)^n*2^(6 n + 3) (Zeta[-2 n - 1, 1/2] - Zeta[-2 n - 1, 1]); Array[f, 15, 0] (* Robert G. Wilson v, Mar 11 2015 *)
max = 20; Clear[g]; g[max + 2] = 1; g[k_] := g[k] = 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/g[k+1]); gf = g[0]; CoefficientList[Series[gf, {x, 0, max}], x] (* Vaclav Kotesovec, Jun 01 2015, after Sergei N. Gladkovskii *)

a(n) = (-1)^n*2^(4*n+1)*(E(2*n+1,1/2)-E(2*n+1,0)), where E(n,x) are the Euler polynomials.
a(n) = A000825(2*n+1).
a(n) = A000828(2*n+1).
a(n) = A000831(2*n+1)/2.
a(n) = A012393(n+1)/2.
G.f.: S(0), where S(k)= 1 - 4*x*(k+1)*(k+2)/(4*x*(k+1)*(k+2) - 1/S(k+1)); (continued fraction). - Sergei N. Gladkovskii, May 28 2015
a(n) ~ (2*n+1)! * 2^(4*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Jun 01 2015

cp's OEIS Frontend

A000831 Expansion of e.g.f. (1 + tan(x))/(1 - tan(x)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

PARI

Sage

SageMath

Formula

A000825 Expansion of cos x (1 + sin x ) /cos 2x.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A092635 Coefficients in asymptotic expansion of (Pi*n/16^n)*binomial(2n,n)^2 in powers of 1/(8n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

PARI

Formula

A253165 a(n) = (-1)^n*2^(6*n+3)*(zeta(-2*n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

Formula

A092635 Coefficients in asymptotic expansion of (Pin/16^n)binomial(2n,n)^2 in powers of 1/(8n).

A253165 a(n) = (-1)^n2^(6n+3)(zeta(-2n-1,1/2) - zeta(-2*n-1,1)), where zeta(a,z) is the generalized Riemann zeta function.