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.

A245166 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (cosh(2*x) - sinh(2*x)*cosh(3*x)).

Original entry on oeis.org

1, 5, 25, 215, 2425, 33875, 569125, 11160035, 250047025, 6302723075, 176522216125, 5438291613155, 182773714292425, 6654680279353475, 260930805319957525, 10961922511422743075, 491220886240696086625, 23388149451193115459075, 1179066988050425638569325
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 + 5*x + 25*x^2/2! + 215*x^3/3! + 2425*x^4/4! + 33875*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + 2*x + 4*x^2/2! + 62*x^3/3! + 448*x^4/4! + 5882*x^5/5! +...
C(x) = 1 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
are the e.g.f.s of A245164 and A245165, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 25*x^2/2! + 2425*x^4/4! + 569125*x^6/6! + 250047025*x^8/8! +...
A1(x) = 5*x + 215*x^3/3! + 33875*x^5/5! + 11160035*x^7/7! + 6302723075*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 5*x + 90*x^3/3! + 8250*x^5/5! + 1946910*x^7/7! + 855018450*x^9/9! + 603621345030*x^11/11! + 624997732481850*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245164, A245165, A245140, A245155.
Cf. A322620, A322190.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/(Cosh[2x]-Sinh[2x]Cosh[3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 21 2024 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (cosh(2*X) - sinh(2*X)*cosh(3*X)), n)}
for(n=0, 30, print1(a(n), ", "))
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(2*X) + sinh(2*X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(2*X)) / (1 - sinh(2*X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: (cosh(2*x) + sinh(2*x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(2*x)) / (1 - sinh(2*x)^2*sinh(3*x)^2).
E.g.f.: (cosh(2*x)*cosh(3*x) + sinh(2*x) + sinh(3*x)) / (1 - sinh(2*x)*sinh(3*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 A245164 and A245165, 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(2*x) = (A0(x) + A1(x)*cosh(3*x)) * (cosh(3*x) - sinh(3*x)*A0(x)) / (1 - sinh(3*x)^2*A1(x)^2).
(3) exp(3*x) = (A0(x) + A1(x)*cosh(2*x)) * (cosh(2*x) - sinh(2*x)*A0(x)) / (1 - sinh(2*x)^2*A1(x)^2).
From Paul D. Hanna, Dec 22 2018: (Start)
(4) exp(2*x) = (A0(x)*cosh(3*x) + A1(x) - sinh(3*x)) / (1 + sinh(3*x)*A1(x)).
(5) exp(3*x) = (A0(x)*cosh(2*x) + A1(x) - sinh(2*x)) / (1 + sinh(2*x)*A1(x)). (End)
FORMULAS FOR TERMS.
a(n) == 0 (mod 5) for n>0.
From Paul D. Hanna, Dec 22 2018: (Start)
a(n) = Sum_{k=0..n} 2^(n-k) * 3^k * A322620(n,k).
a(n) = Sum_{k=0..n} 2^(n-k) * 3^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

A245138 E.g.f.: (cosh(x) + sinh(x)*cosh(2*x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2).

Original entry on oeis.org

1, 1, 1, 13, 49, 361, 3121, 39733, 409249, 6410641, 91979041, 1716516253, 29795642449, 660718214521, 13656276138961, 345794520085573, 8290832204163649, 237409681243284001, 6465138777774530881, 206263448435258395693, 6296129943088315156849, 221484543685548532051081
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 + x + x^2/2! + 13*x^3/3! + 49*x^4/4! + 361*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + x^2/2! + 49*x^4/4! + 3121*x^6/6! + 409249*x^8/8! + 91979041*x^10/10! +...
A1(x) = x + 13*x^3/3! + 361*x^5/5! + 39733*x^7/7! + 6410641*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 + 12*x^3/3! + 240*x^5/5! + 24192*x^7/7! + 3452160*x^9/9! + 841961472*x^11/11! + 300389806080*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245139, A245140, A245153, A245164.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(Cosh[x]+Sinh[x]Cosh[2x])/Sqrt[1-Sinh[x]^2 Sinh[2x]^2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 29 2017 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(X) + sinh(X)*cosh(2*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(2*x) - sinh(2*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(2*x)^2), where G(x) is the e.g.f. of A245140.

A245165 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(2*x)) / sqrt(1 - sinh(2*x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 3, 9, 63, 513, 8043, 115209, 2170983, 42235713, 1075192083, 27302385609, 837303386703, 25799446123713, 938330441750523, 34249273199668809, 1436790115786367223, 60444494320614768513, 2873965406506938435363, 137038195324637653852809, 7283819678458854655944543
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 + 3*x + 9*x^2/2! + 63*x^3/3! + 513*x^4/4! + 8043*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 513*x^4/4! + 115209*x^6/6! + 42235713*x^8/8! +...
A1(x) = 3*x + 63*x^3/3! + 8043*x^5/5! + 2170983*x^7/7! + 1075192083*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)) = 3*x + 36*x^3/3! + 4560*x^5/5! + 932736*x^7/7! + 433555200*x^9/9! + 300576731136*x^11/11! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245164, A245166, A245139, A245154.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(Cosh[3x]+Sinh[3x]Cosh[2x])/Sqrt[1-Sinh[ 2x]^2 Sinh[3x]^2],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 27 2021 *)
  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(3*X) + sinh(3*X)*cosh(2*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(2*x) - sinh(2*x)*cosh(3*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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