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.

Previous Showing 11-13 of 13 results.

A218308 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(3*x)*cosh(3*x).

Original entry on oeis.org

1, 3, 42, 1116, 44616, 2394288, 161719200, 13187258304, 1261037553792, 138415816348416, 17155627044653568, 2370099000682257408, 361171910376568571904, 60185513513709805350912, 10887989148395358662270976, 2125192867898778619536457728
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 3*x + 42*x^2/2! + 1116*x^3/3! + 44616*x^4/4! + 2394288*x^5/5! +...
where
A(x) = cosh(3*x) + 3*7^0*cosh(7*x)*x + 3*11^1*cosh(11*x)*x^2/2! + 3*15^2*cosh(15*x)*x^3/3! + 3*19^3*cosh(19*x)*x^4/4! + 3*23^4*cosh(23*x)*x^5/5! +...

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218309, A218310.

Programs

  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*X)))); Egf=exp(3*R)*cosh(3*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,3*(4*k+3)^(k-1)*cosh((4*k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} (4*n+1)^(n-1) * cosh((4*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/4 * LambertW(-4*x * exp(4*x)) ).
a(n) = 3/2 * Sum_{k=0..n} (4*k+3)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 3/2 * Sum_{k>=0} (4*k+3)^(k-1) * x^k/(1 - (4*k+3)*x)^(k+1). (End)

A218309 E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(4*x)*cosh(4*x).

Original entry on oeis.org

1, 4, 56, 1264, 40640, 1711744, 89533184, 5607463936, 409621790720, 34218229227520, 3219000547131392, 336858779869020160, 38823224436435845120, 4886982191317154529280, 667188807538423365632000, 98200163047169655115350016, 15501781660715229538766815232
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + 4*x + 56*x^2/2! + 1264*x^3/3! + 40640*x^4/4! + 1711744*x^5/5! +...
where
A(x) = cosh(2*x) + 4*5^0*cosh(5*x)*x + 4*8^1*cosh(8*x)*x^2/2! + 4*11^2*cosh(11*x)*x^3/3! + 4*14^3*cosh(14*x)*x^4/4! + 4*17^4*cosh(17*x)*x^5/5! +...

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218310.

Programs

  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(3*X)*cosh(3*X)))); Egf=exp(4*R)*cosh(4*R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,4*(3*k+4)^(k-1)*cosh((3*k+4)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} 4*(3*n+4)^(n-1) * cosh((3*n+4)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( 4*x - 4/3 * LambertW(-3*x * exp(3*x)) ).
a(n) = 2 * Sum_{k=0..n} (3*k+4)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 2 * Sum_{k>=0} (3*k+4)^(k-1) * x^k/(1 - (3*k+4)*x)^(k+1). (End)

A218310 E.g.f. A(x) satisfies A( x/(exp(5*x)*cosh(5*x)) ) = exp(x)*cosh(x).

Original entry on oeis.org

1, 1, 12, 364, 17248, 1118816, 92306432, 9251542784, 1091729307648, 148280571406336, 22785577791987712, 3908379504145178624, 740274425760340901888, 153456630172316832628736, 34557831428406144298647552, 8401098284435734877893033984
Offset: 0

Views

Author

Paul D. Hanna, Oct 25 2012

Keywords

Comments

More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.

Examples

			E.g.f.: A(x) = 1 + x + 12*x^2/2! + 364*x^3/3! + 17248*x^4/4! + 1118816*x^5/5! +...
where
A(x) = cosh(x) + 6^0*cosh(6*x)*x + 11^1*cosh(11*x)*x^2/2! + 16^2*cosh(16*x)*x^3/3! + 21^3*cosh(21*x)*x^4/4! + 26^4*cosh(26*x)*x^5/5! +...

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218309.

Programs

  • PARI
    {a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(5*X)*cosh(5*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))
  • PARI
    /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1,X=x+x*O(x^n)); Egf=sum(k=0,n,(5*k+1)^(k-1)*cosh((5*k+1)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} (5*n+1)^(n-1) * cosh((5*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/5 * LambertW(-5*x * exp(5*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(1 - (5*k+1)*x)^(k+1). (End)
Previous Showing 11-13 of 13 results.

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