cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A380056 E.g.f. (exp(x) - 1)/cos(2*x).

Original entry on oeis.org

1, 1, 13, 25, 441, 1261, 30213, 115025, 3529201, 16792021, 629401213, 3593565625, 159175688361, 1060279600381, 54189700721013, 412526870321825, 23894940183997921, 204641610743378341, 13248060325188261613, 126065945039257743625, 9020317522757414377881, 94419130586604915837901
Offset: 1

Views

Author

Paul D. Hanna, Jan 28 2025

Keywords

Comments

Conjecture: a(n) == 1 (mod 4) for n >= 1.
Conjecture: a(4*n) is divisible by 5 for n >= 1.

Examples

			E.g.f.: A(x) = x + x^2/2! + 13*x^3/3! + 25*x^4/4! + 441*x^5/5! + 1261*x^6/6! + 30213*x^7/7! + 115025*x^8/8! + 3529201*x^9/9! + 16792021*x^10/10! + ...

Links

Crossrefs

Cf. A380053, A000364, A380555.

Programs

  • PARI
    {a(n) = my(A,X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(2*X), n)}
for(n=1,25,print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = (exp(x) - 1)/cos(2*x).
(2) A(F(x)) = x where F(x) = log(1 + x*cos(2*F(x))) equals the e.g.f. of A380555.
(3.a) a(2*n+1) = Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(n-k) * 4^(n-k) for n >= 0.
(3.b) a(2*n) = Sum_{k=1..n} binomial(2*n, 2*k) * A000364(n-k) * 4^(n-k) for n >= 1.
a(n) ~ (exp(Pi/4) - 1 + (exp(-Pi/4) - 1)*(-1)^n) * 2^(2*n + 3/2) * n^(n + 1/2) / (sqrt(Pi) * exp(n) * Pi^n). - Vaclav Kotesovec, Jan 29 2025

A380556 E.g.f. A(x) satisfies A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.

Original entry on oeis.org

1, 1, 0, -12, 48, 820, -14160, -69160, 5900160, -44796960, -3089865600, 88646729600, 1412786918400, -135956951062400, 1023512450688000, 203887248898944000, -7307555382586368000, -252816835499795840000, 26110132266648748032000, -95216226972043640320000, -80962066973581160140800000
Offset: 0

Views

Author

Paul D. Hanna, Jan 28 2025

Keywords

Examples

			E.g.f.: A(x) = 1 + x - 12*x^3/3! + 48*x^4/4! + 820*x^5/5! - 14160*x^6/6! - 69160*x^7/7! + 5900160*x^8/8! - 44796960*x^9/9! - 3089865600*x^10/10! + ...
 where A(x) = 1 + x*cos( log( A(x)^2 ) ).
RELATED SERIES.
A(x)^(2*i) = 1 + 2*i*x + (-4 - 2*i)*x^2/2! + (12 - 28*i)*x^3/3! + (164 + 228*i)*x^4/4! + (-2360 + 1440*i)*x^5/5! + (-9880 - 51480*i)*x^6/6! + (737520 + 129440*i)*x^7/7! + (-4977440 + 17821440*i)*x^8/8! + (-308986560 - 306791360*i)*x^9/9! + ...
 where A(x)^(2*i) = cos( log(A(x)^2) ) + i*sin( log(A(x)^2) ).
L(x) = log(A(x)) = x - x^2 - 10*x^3 + 90*x^4 + 364*x^5 - 17760*x^6 + 85280*x^7 + 5447120*x^8 + ... + A380555(n)*x^n/n! + ...
 where L( (exp(x) - 1)/cos(2*x) ) = x.
cos(2*L(x)) = 1 - 4*x^2/2! + 12*x^3/3! + 164*x^4/4! - 2360*x^5/5! - 9880*x^6/6! + 737520*x^7/7! - 4977440*x^8/8! + ... + (a(n+1)/(n+1))*x^n/n! + ...
 where A(x) = 1 + x*cos(2*L(x)).

Crossrefs

Cf. A380057, A380555.

Programs

  • Mathematica
    CoefficientList[Exp[InverseSeries[Series[ (Exp[x] - 1)/Cos[2*x] ,{x,0,20}],x]],x]Range[0,20]! (* Stefano Spezia, Jan 29 2025 *)
  • PARI
    {a(n) = my(A = 1+x+x*O(x^n)); for(i=1,n, A = real( 1 + x*A^(2*I) +x*O(x^n) )); n!*polcoef(A,n)}
for(n=0,30, print1(a(n),", "))

Formula

E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.
(1) A(x) = real( 1 + x*A(x)^(2*i) ), where i^2 = -1.
(2) A(x) = 1 + x*cos( log( A(x)^2 ) ).
(3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(2*x) ) is the e.g.f. of A380555.
(4) A( (exp(x) - 1)/cos(2*x) ) = exp(x).
Showing 1-2 of 2 results.

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