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.

A245155 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / (cosh(x) - sinh(x)*cosh(3*x)).

Original entry on oeis.org

1, 4, 16, 100, 832, 8644, 107776, 1567780, 26063872, 487466884, 10129985536, 231560895460, 5774444019712, 155997355725124, 4538464905527296, 141469868440031140, 4703786933664612352, 166172927821116399364, 6215792183431115309056, 245422172388559255422820
Offset: 0

Views

Author

Paul D. Hanna, Jul 12 2014

Keywords

Comments

Limit (a(n)/n!)^(-1/n) = log(t) = 0.4812118250596... where t = (1+sqrt(5))/2 satisfies 1 + t + t^3 = t^4.

Examples

			E.g.f.: A(x) = 1 + 4*x + 16*x^2/2! + 100*x^3/3! + 832*x^4/4! + 8644*x^5/5! +...
such that A(x) = B(x)*C(x), where
B(x) = 1 + x + x^2/2! + 28*x^3/3! + 109*x^4/4! + 1036*x^5/5! + 12421*x^6/6! +...
C(x) = 1 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! + 26109*x^6/6! +...
are the e.g.f.s of A245153 and A245154, respectively.
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 16*x^2/2! + 832*x^4/4! + 107776*x^6/6! + 26063872*x^8/8! +...
A1(x) = 4*x + 100*x^3/3! + 8644*x^5/5! + 1567780*x^7/7! + 487466884*x^9/9! +...
then A0(x)^2 - A1(x)^2 = 1.
Note that the logarithm is an odd function:
log(A(x)) = 4*x + 36*x^3/3! + 1860*x^5/5! + 240996*x^7/7! + 58280580*x^9/9! + 22651336356*x^11/11! + 12912049359300*x^13/13! + 10148316042271716*x^15/15! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245153, A245154, A245140, A245166.
Cf. A322620, A322190.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff( (cosh(3*X) + sinh(3*X)*cosh(X)) / (cosh(X) - sinh(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(X) + sinh(X)*cosh(3*X)) * (cosh(3*X) + sinh(3*X)*cosh(X)) / (1 - sinh(X)^2*sinh(3*X)^2), n)}
for(n=0, 30, print1(a(n), ", "))

Formula

E.g.f.: (cosh(x) + sinh(x)*cosh(3*x)) * (cosh(3*x) + sinh(3*x)*cosh(x)) / (1 - sinh(x)^2*sinh(3*x)^2).
E.g.f.: (cosh(x)*cosh(3*x) + sinh(x) + sinh(3*x)) / (1 - sinh(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 A245153 and A245154, 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(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(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(3*x) + A1(x) - sinh(3*x)) / (1 + sinh(3*x)*A1(x)).
(5) exp(3*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} 3^k * A322620(n,k).
a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * A322190(n,k). (End)
a(n) ~ 2*sqrt(2*Pi/5) * n^(n+1/2) / (exp(n) * (log((1+sqrt(5))/2))^(n+1)). - 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.

A245154 E.g.f.: (cosh(3*x) + sinh(3*x)*cosh(x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2).

Original entry on oeis.org

1, 3, 9, 36, 189, 2148, 26109, 371136, 5407929, 95795568, 1832049009, 41428038336, 972380766069, 25736128903488, 705111069908709, 21600790506395136, 683861855417706609, 23836956839153265408, 853476673589938069209, 33263825890074489025536
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 + 3*x + 9*x^2/2! + 36*x^3/3! + 189*x^4/4! + 2148*x^5/5! +...
Let A(x) = A0(x) + A1(x) where
A0(x) = 1 + 9*x^2/2! + 189*x^4/4! + 26109*x^6/6! + 5407929*x^8/8! +...
A1(x) = 3*x + 36*x^3/3! + 2148*x^5/5! + 371136*x^7/7! + 95795568*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 + 9*x^3/3! + 1095*x^5/5! + 119469*x^7/7! + 28399275*x^9/9! + 11494484529*x^11/11! + 6432743099055*x^13/13! +...
thus A(x)*A(-x) = 1.

Crossrefs

Cf. A245153, A245155, A245139, A245165.

Programs

  • PARI
    {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff((cosh(3*X) + sinh(3*X)*cosh(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(x) - sinh(x)*cosh(3*x)) / sqrt(1 - sinh(x)^2*sinh(3*x)^2), where G(x) is the e.g.f. of A245155.
a(n) ~ 2*sqrt(2) * n^n / (5^(1/4) * exp(n) * (log((1+sqrt(5))/2))^(n+1/2)). - Vaclav Kotesovec, Nov 04 2014

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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