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 1-4 of 4 results.

A245140 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(2*x)).

Original entry on oeis.org

1, 3, 9, 45, 297, 2433, 23949, 275145, 3612177, 53348193, 875453589, 15802999545, 311196040857, 6638817262353, 152521855979229, 3754366520240745, 98575724288354337, 2749997026637342913, 81230299711952152869, 2532707187355266614745, 83124358113443446120617, 2864579803637260793877873
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.
In general, the radius of convergence r of the e.g.f. (cosh(p*x) + sinh(p*x)*cosh(q*x)) / (cosh(q*x) - sinh(q*x)*cosh(p*x)), where p and q are positive integers, equals r = log(t) such that t is the positive real root that satisfies: 1 + t^p + t^q = t^(p+q).

Examples

			E.g.f.: A(x) = 1 + 3*x + 9*x^2/2! + 45*x^3/3! + 297*x^4/4! + 2433*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! + 3121*x^6/6! +...
C(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! + 5344*x^6/6! +...
are the e.g.f.s of A245138 and A245139, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 297*x^4/4! + 23949*x^6/6! + 3612177*x^8/8! +...
A1(x) = 3*x + 45*x^3/3! + 2433*x^5/5! + 275145*x^7/7! + 53348193*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 3*x + 18*x^3/3! + 570*x^5/5! + 46158*x^7/7! + 6959250*x^9/9! + 1686709398*x^11/11! + 599570355930*x^13/13! + 3754366520240745*x^15/15! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245138, A245139, A245155, A245166.
Cf. A322620, A322190.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(2*X) + sinh(2*X)*cosh(X)) / (cosh(X) - sinh(X)*cosh(2*X)), n)}
for(n=0,30,print1(a(n),", "))
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(2*X)) * (cosh(2*X) + sinh(2*X)*cosh(X)) / (1 - sinh(X)^2*sinh(2*X)^2),n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) * (cosh(2*x) + sinh(2*x)*cosh(x)) / (1 - sinh(x)^2*sinh(2*x)^2).
E.g.f.: (cosh(x)*cosh(2*x) + sinh(x) + sinh(2*x)) / (1 - sinh(x)*sinh(2*x)). - Paul D. Hanna, Dec 22 2018
E.g.f.: A(x) = B(x)*C(x), where B(x) and C(x) are the e.g.f.s of A245138 and A245139, respectively.
Let e.g.f. A(x) = A0(x) + A1(x) where A0(x) = (A(x)+A(-x))/2 and A1(x) = (A(x)-A(-x))/2, then:
(1) A0(x)^2 - A1(x)^2 = 1.
(2) exp(x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).
(3) exp(2*x) = (A0(x) + A1(x)*cosh(x)) * (cosh(x) - sinh(x)*A0(x)) / (1 - sinh(x)^2*A1(x)^2).
From Paul D. Hanna, Dec 22 2018: (Start)
(4) exp(x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)).
(5) exp(2*x) = (A0(x)*cosh(x) + A1(x) - sinh(x)) / (1 + sinh(x)*A1(x)). (End)
FORMULAS FOR TERMS.
From Paul D. Hanna, Dec 22 2018: (Start)
a(n) = Sum_{k=0..n} 2^k * A322620(n,k).
a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * A322190(n,k). (End)

A245153 E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 1, 1, 28, 109, 1036, 12421, 189568, 2377369, 50888656, 889772041, 21056972608, 463426778629, 13171920918976, 338302052475661, 11024635871323648, 331174000888419889, 12111179923298826496, 413871819030803915281, 16886967133601994738688
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log( (1+sqrt(5))/2 ) = 0.4812118250596...

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + x^2/2! + 109*x^4/4! + 12421*x^6/6! + 2377369*x^8/8! +...
A1(x) = x + 28*x^3/3! + 1036*x^5/5! + 189568*x^7/7! + 50888656*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = x + 27*x^3/3! + 765*x^5/5! + 121527*x^7/7! + 29881305*x^9/9! + 11156851827*x^11/11! + 6479306260245*x^13/13! +...
thus A(x)*A(-x) = 1.

Links

Crossrefs

Cf. A245154, A245155, A245138, A245164.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[3x])/Sqrt[1-Sinh[ x]^2 Sinh[3x]^2 ],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2020 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(3*X)) / sqrt(1 - sinh(X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: G(x) * (cosh(3*x) - sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014

A245139 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2).

Original entry on oeis.org

1, 2, 4, 14, 64, 602, 5344, 58214, 661504, 9666482, 145897984, 2611988414, 47548524544, 1002692887562, 21581168410624, 527328466446614, 13084553110749184, 362312592419199842, 10175324275879051264, 315223836841156264814, 9889646730551557095424, 338833067799589889659322
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.609377863436... where t is the tribonacci constant and satisfies 1 + t + t^2 = t^3.

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 64*x^4/4! + 602*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 4*x^2/2! + 64*x^4/4! + 5344*x^6/6! + 661504*x^8/8! +...
A1(x) = 2*x + 14*x^3/3! + 602*x^5/5! + 58214*x^7/7! + 9666482*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = 2*x + 6*x^3/3! + 330*x^5/5! + 21966*x^7/7! + 3507090*x^9/9! + 844747926*x^11/11! + 299180549850*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245138, A245140, A245154, A245165.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(X)) / sqrt(1 - sinh(X)^2*sinh(2*X)^2),n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f.: G(x) * (cosh(x) - sinh(x)*cosh(2*x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2), where G(x) is the e.g.f. of A245140.

A245164 E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 2, 4, 62, 448, 5882, 82144, 1762742, 32401408, 839773682, 20709251584, 658128799022, 19691428538368, 735018387765482, 26206768383361024, 1124046915311796902, 46319665594721763328, 2246606049886763789282, 105187723831561379774464, 5688928855528010885284382
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.3570506972213... where t satisfies 1 + t^2 + t^3 = t^5.

Examples

			E.g.f.: A(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 4*x^2/2! + 448*x^4/4! + 82144*x^6/6! + 32401408*x^8/8! +...
A1(x) = 2*x + 62*x^3/3! + 5882*x^5/5! + 1762742*x^7/7! + 839773682*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm of the e.g.f. is an odd function:
Log(A(x)) = 2*x + 54*x^3/3! + 3690*x^5/5! + 1014174*x^7/7! + 421463250*x^9/9! + 303044613894*x^11/11! + 312200620305210*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245165, A245166, A245138, A245153.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(Cosh[2x]+Sinh[2x]Cosh[3x])/Sqrt[1- Sinh[ 2x]^2 Sinh[3x]^2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 01 2016 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) / sqrt(1 - sinh(2*X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: G(x) * (cosh(3*x) - sinh(3*x)*cosh(2*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245166.
Showing 1-4 of 4 results.

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