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 1-3 of 3 results.

A380053 E.g.f. (exp(x) - 1)/cos(x).

Original entry on oeis.org

1, 1, 4, 7, 36, 91, 624, 2087, 18256, 76231, 814144, 4078867, 51475776, 300870571, 4381112064, 29265244847, 482962852096, 3629392540111, 66942218896384, 558956224522027, 11394877025289216, 104659828714136851, 2336793875186479104, 23414201065072302407, 568240131312188379136
Offset: 1

Views

Author

Paul D. Hanna, Jan 24 2025

Keywords

Examples

			E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + 18256*x^9/9! + 76231*x^10/10! + ...
RELATED SERIES.
1/cos(x) = 1 + x^2/2! + 5*x^4/4! + 61*x^6/6! + 1385*x^8/8! + 50521*x^10/10! + 2702765*x^12/12! + 199360981*x^14/14! + ... + A000364(n)*x^(2*n)/(2*n)! + ...
Let F(x) be the series reversion of e.g.f. A(x) so that A(F(x)) = x, then
F(x) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + ... + A380055(n)*x^n/n! + ...
where F(x) = log(1 + x*cos(F(x))).

Links

Crossrefs

Cf. A002084, A000364, A380054, A380055, A380057.

Programs

  • PARI
    {a(n) = my(X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(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(x).
(2) A(F(x)) = x where F(x) = log(1 + x*cos(F(x))) equals the e.g.f. of A380055.
(3.a) a(2*n+1) = A002084(n) for n >= 0.
(3.b) a(2*n+1) = Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(n-k) for n >= 0.
(3.c) a(2*n) = Sum_{k=1..n} binomial(2*n, 2*k) * A000364(n-k) for n >= 1.
a(n) ~ (-1 + exp(Pi/2) + (-1 + exp(-Pi/2))*(-1)^n) * 2^(n + 3/2) * n^(n + 1/2) / (exp(n) * Pi^(n + 1/2)). - Vaclav Kotesovec, Jan 24 2025

A380057 E.g.f. satisfies A(x) = real( 1 + x*A(x)^i ), where i^2 = -1.

Original entry on oeis.org

1, 1, 0, -3, 12, 10, -660, 5600, 8400, -951660, 12715200, 21635900, -4308744000, 80314007800, 115204471200, -44501789202000, 1083368456352000, 782537744170000, -876176569052928000, 26724653123017850000, -10930955906482560000, -29304692085200613900000, 1088420125090964265600000
Offset: 0

Views

Author

Paul D. Hanna, Jan 24 2025

Keywords

Examples

			E.g.f.: A(x) = 1 + x - 3*x^3/3! + 12*x^4/4! + 10*x^5/5! - 660*x^6/6! + 5600*x^7/7! + 8400*x^8/8! - 951660*x^9/9! + 12715200*x^10/10! + ...
 where A(x) = real( 1 + x*A(x)^i ).
RELATED SERIES.
A(x)^i = 1 + i*x + (-1 - i)*x^2/2! + (3 - 2*i)*x^3/3! + (2 + 24*i)*x^4/4! + (-110 - 90*i)*x^5/5! + (800 - 540*i)*x^6/6! + (1050 + 12640*i)*x^7/7! + (-105740 - 85680*i)*x^8/8! + (1271520 - 808480*i)*x^9/9! + ...
L(x) = log(A(x)) = x - x^2/2! - x^3/3! + 18*x^4/4! - 86*x^5/5! - 210*x^6/6! + 8840*x^7/7! - 80080*x^8/8! - 266220*x^9/9! + ... + A380055(n)*x^n/n! + ...
 where L( (exp(x) - 1)/cos(x) ) = x.
cos(L(x)) = 1 - x^2/2! + 3*x^3/3! + 2*x^4/4! - 110*x^5/5! + 800*x^6/6! + 1050*x^7/7! - 105740*x^8/8! + 1271520*x^9/9! + ... + (a(n+1)/(n+1))*x^n/n! + ...
 where A(x) = 1 + x*cos(L(x)).
sin(L(x)) = x - x^2/2! - 2*x^3/3! + 24*x^4/4! - 90*x^5/5! - 540*x^6/6! + 12640*x^7/7! - 85680*x^8/8! - 808480*x^9/9! + 29636100*x^10/10! + ...
 where A(x)^i = cos(L(x)) + i*sin(L(x)).
(exp(x) - 1)/cos(x) = x + x^2/2! + 4*x^3/3! + 7*x^4/4! + 36*x^5/5! + 91*x^6/6! + 624*x^7/7! + 2087*x^8/8! + 18256*x^9/9! + ... + A380053(n)*x^n/n! + ...
 where A( (exp(x) - 1)/cos(x) ) = exp(x).
SPECIFIC VALUES.
A(t) = 3/2 at t = 0.544117577471602714300218052469...
  where t = (1/2)/real( (3/2)^i ) = (1/2)/cos(log(3/2)).
A(t) = 4/3 at t = 0.347619044845549508650339299345...
  where t = (1/3)/cos(log(4/3)).
A(t) = 5/4 at t = 0.256355931938414599112107620645...
  where t = (1/4)/cos(log(5/4)).
A(t) = 6/5 at t = 0.203370786475336228292002468204...
  where t = (1/5)/cos(log(6/5)).
A(5/9) = 1.509166099041466745516450953830986339123819257989853587...
A(1/2) = 1.464101647089238181480821477030380839753029025795281424...
  where A(1/2) = real(1 + (1/2)*A(1/2)^i) = 1 + (1/2)*cos(log( A(1/2) )).
A(1/3) = 1.320532220599397916493026637544832197337142648617761054...
  where A(1/3) = 1 + (1/3)*cos(log( A(1/3) )).
A(1/4) = 1.244062312309475342914040842626056476267801020397804125...
  where A(1/4) = 1 + (1/4)*cos(log( A(1/4) )).
A(1/5) = 1.196781755871960043253050643301603078428252760913031148...
A(1/6) = 1.164732612936027556202815971272406070730644075186904648...
A(1/10) = 1.099550027586687118480790742605384041459169163472092691...

Links

Crossrefs

Cf. A380053, A380055, A380058.

Programs

  • PARI
    {a(n) = my(A = 1+O(x)); for(i=0, n, A = real( 1 + x*A^I )); 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)^i ), where i^2 = -1.
(2) A(x) = 1 + x*cos( log( A(x) ) ).
(3) A(x) = exp( L(x) ), where L(x) = Series_Reversion( (exp(x) - 1)/cos(x) ) is the e.g.f. of A380055.
(4) A( (exp(x) - 1)/cos(x) ) = exp(x).
(5) A'(x) = A(x) * cos( log(A(x)) ) / (A(x) + x*sin( log(A(x)) )).

A380555 E.g.f. A(x) satisfies A(x) = log( 1 + x * cos(2*A(x)) ).

Original entry on oeis.org

1, -1, -10, 90, 364, -17760, 85280, 5447120, -116082720, -1709304480, 123520217600, -637137072000, -136024779843200, 3988924415257600, 131963952741504000, -11250603940363008000, 19125068757338752000, 28119635304260378112000, -943657308179458552576000, -59184868918118854443520000
Offset: 1

Views

Author

Paul D. Hanna, Jan 28 2025

Keywords

Examples

			E.g.f.: A(x) = x - x^2/2! - 10*x^3/3! + 90*x^4/4! + 364*x^5/5! - 17760*x^6/6! + 85280*x^7/7! + 5447120*x^8/8! - 116082720*x^9/9! - 1709304480*x^10/10! + ...
SPECIAL VALUES.
A(t) = Pi/16 at t = 0.2348273750777024091348769029539035346094...
  where t = (exp(Pi/16) - 1) * 2/sqrt(2 + sqrt(2)).
A(t) = Pi/20 at t = 0.1788419348189777972181354090557549056970...
  where t = (exp(Pi/20) - 1) * sqrt(2/5) * sqrt(5 - sqrt(5)).
A(t) = Pi/24 at t = 0.1447869365509419179517924812606040896260...
  where t = (exp(Pi/24) - 1) * sqrt(2) * (sqrt(3) - 1).
A(t) = Pi/32 at t = 0.1051765042663303122710070527373480540972...
  where t = (exp(Pi/32) - 1) * 2/sqrt(2 + sqrt(2 + sqrt(2))).
SPECIFIC VALUES.
A(1/4) = 0.20622896305658178490114810132496023364946226486284...
  where A(1/4) = log( 1 + (1/4)*cos(2*A(1/4)) ).
A(1/5) = 0.17245713659793550611733876887712582250401979882536...
  where A(1/5) = log( 1 + (1/5)*cos(2*A(1/5)) ).
A(1/6) = 0.14792487411803287676006534562611718490228530287793...
  where A(1/6) = log( 1 + (1/6)*cos(2*A(1/6)) ).
A(1/8) = 0.11485990457955157002021678730408576231050832334885...
  where A(1/8) = log( 1 + (1/8)*cos(2*A(1/8)) ).

Links

Crossrefs

Cf. A380055, A380556.

Programs

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

Formula

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

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