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.

A215368 E.g.f.: Series_Reversion( x*cos(x) - x*sin(x) ).

Original entry on oeis.org

1, 2, 15, 176, 2905, 61536, 1592703, 48706048, 1718376561, 68702272000, 3069734553743, 151592011714560, 8198710703202825, 481965222651551744, 30598546651134134655, 2086474763912627879936, 152083996930329322871521, 11800530001358902191587328, 971113004536128839898536079
Offset: 1

Views

Author

Paul D. Hanna, Aug 08 2012

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 15*x^3/3! + 176*x^4/4! + 2905*x^5/5! +...
where A(x*cos(x) - x*sin(x)) = x and A(x) = x/(cos(A(x)) - sin(A(x))).
Related expansions:
cos(A(x)) = 1 - x^2/2! - 6*x^3/3! - 71*x^4/4! - 1160*x^5/5! - 24481*x^6/6! - 631904*x^7/7! - 19288079*x^8/8! -...
sin(A(x)) = x + 2*x^2/2! + 14*x^3/3! + 164*x^4/4! + 2696*x^5/5! + 57006*x^6/6! + 1473632*x^7/7! + 45026344*x^8/8! +...

Links

Crossrefs

Cf. A201923, A215638.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n));n!*polcoeff(serreverse(x*cos(X)-x*sin(X)),n)}
  • PARI
    {a(n)=local(X=x+x^2*O(x^n));n!*polcoeff(x/(cos(X)-sin(X))^n/n,n)}
for(n=1,31,print1(a(n),", "))

Formula

E.g.f. satisfies: A(x) = x / (cos(A(x)) - sin(A(x))).
a(n) = [x^n/n!] 1/(cos(x)-sin(x))^n / n.
a(n) = n*A201923(n-1).
a(n) ~ sqrt(-1 + 4/(3 + sin(2*s))) * n^(n-1) / (r^n * exp(n)), where s = 0.4026281741881116098199325239112307245635064777960... is the root of the equation s*cos(2*s) + sin(2*s) = 1 and r = s*(cos(s) - sin(s)) = 0.21266685344074710045360679397024815598865409988038310855608986167... - Vaclav Kotesovec, Oct 04 2020

A215639 E.g.f. satisfies: A(x) = cos(x*A(x)) + sin(x*A(x)).

Original entry on oeis.org

1, 1, 1, -4, -51, -304, 125, 34880, 557753, 3416320, -74779911, -2917151744, -46015368443, 115191402496, 30429734385973, 942941062774784, 9925460231059185, -471696770041053184, -29508689065235461903, -733077456673636089856, 4714209123766494329021
Offset: 0

Views

Author

Paul D. Hanna, Aug 18 2012

Keywords

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! - 4*x^3/3! - 51*x^4/4! - 304*x^5/5! + 125*x^6/6! +...
where A(x) = cos(x*A(x)) + sin(x*A(x)).
Related expansions:
cos(x*A(x)) = 1 - x^2/2! - 6*x^3/3! - 23*x^4/4! + 40*x^5/5! + 2159*x^6/6! + 26656*x^7/7! + 114577*x^8/8! +...
sin(x*A(x)) = x + 2*x^2/2! + 2*x^3/3! - 28*x^4/4! - 344*x^5/5! - 2034*x^6/6! + 8224*x^7/7! + 443176*x^8/8! +...

Crossrefs

Cf. A215638, A201923.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(1/x*serreverse(x/(cos(X)+sin(X))), n)}
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cos(X)+sin(X))^(n+1)/(n+1), n)}
  • PARI
    {a(n)=local(A=1+x+x^2*O(x^n)); for(i=1,n,A=cos(x*A)+sin(x*A));n!*polcoeff(A, n)}
for(n=0, 31, print1(a(n), ", "))

Formula

E.g.f.: A(x) = (1/x)*Series_Reversion( x/(cos(x) + sin(x)) ).
E.g.f. satisfies: A(x/(cos(x) + sin(x))) = cos(x) + sin(x).
a(n) = [x^n/n!] (cos(x)+sin(x))^(n+1) / (n+1).
a(n) = A215638(n+1)/(n+1) for n>=0.

A201990 E.g.f. satisfies: A(x) = 1/(cos(x*A(x)^2) - sin(x*A(x)^2)).

Original entry on oeis.org

1, 1, 7, 95, 1969, 55201, 1956375, 83935039, 4230528353, 245059707841, 16043680004903, 1171567218325151, 94415150206330641, 8323801562833775201, 796927800013656980791, 82342529545666235490431, 9132868398860301753027265, 1082287792241161814647419265
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2011

Keywords

Comments

Compare e.g.f. to: Sum_{n>=0} (2*n+1)^(n-1)*x^n/n! = sqrt((1/x)*Series_Reversion(x*(cosh(x) - sinh(x))^2)).
The radius of convergence r of e.g.f. A(x) is given by:
r = t*(cos(t) - sin(t))^2 where t = (1 - sin(2*t))/(2*cos(2*t)), so that:
r = 0.13127 35638 55724 99317 13322 82818 86189 50670 52604 32023 ...
t = 0.27798 42153 59698 32056 15352 87789 00442 74782 64480 84947 ...
Further, A(r) = 1/(cos(t) - sin(t)), thus
A(r) = 1.45519 57921 91350 02891 97122 64456 17664 48847 98244 19461 ...

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 95*x^3/3! + 1969*x^4/4! + 55201*x^5/5! +...
where
1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! + 24611*x^7/7! +...+ A001586(n)*x^n/n! +...
The coefficients of x^n/n! in odd powers of G(x) = 1/(cos(x)-sin(x)) begin:
G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
G^3: [1,(3), 15, 93, 705, 6243, 63375, 724413, ...];
G^5: [1, 5,(35), 295, 2905, 32525, 407435, 5638495, ...];
G^7: [1, 7, 63,(665), 8001, 107527, 1592703, 25738265, ...];
G^9: [1, 9, 99, 1251, (17721), 276849, 4716459, 86873211, ...];
G^11:[1, 11, 143, 2101, 34177, (607211), 11668943, 240764821, ...];
G^13:[1, 13, 195, 3263, 59865, 1190293,(25432875), 580193783, ...];
G^15:[1, 15, 255, 4785, 97665, 2146575, 50429055,(1259025585), ...]; ...
where coefficients in parenthesis form the initial terms of this sequence:
[1/1, 3/3, 35/5, 665/7, 17721/9, 607211/11, 25432875/13, 1259025585/15, ...].

Crossrefs

Cf. A201923, A001586.

Programs

  • Mathematica
    CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x])^2, {x, 0, 21}], x]],x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n));n!*polcoeff(sqrt(1/x*serreverse(x*(cos(X)-sin(X))^2)),n)}
  • PARI
    {a(n)=local(A=1+x,X=x+x*O(x^n));for(i=1,n,A=1/(cos(X*A^2) - sin(X*A^2)));n!*polcoeff(A,n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n),A001586=1/(cos(X)-sin(X)));n!*polcoeff(A001586^(2*n+1),n)/(2*n+1)}

Formula

E.g.f. satisfies: A( x*(cos(x) - sin(x))^2 ) = 1/(cos(x) - sin(x)).
E.g.f: sqrt( (1/x) * Series_Reversion( x*(cos(x) - sin(x))^2 ) ).
a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(2*n+1) / (2*n+1).
a(n) ~ sqrt((t*cos(2*t))/(2*cos(2*t)+4*t*(3+sin(2*t)))) * n^(n-1) / (exp(n) * r^(n+1/2)), where r and t were described above. - Vaclav Kotesovec, Jan 12 2014
Showing 1-3 of 3 results.

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