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 1-5 of 5 results.

A258922 E.g.f. satisfies A(x) = 1/(3 - 2*exp(x*A(x))).

Original entry on oeis.org

1, 2, 18, 302, 7562, 253542, 10685794, 543309230, 32378850042, 2214215333750, 170939286647570, 14707184259036414, 1395561779648175274, 144795755972202587462, 16308198003201872476866, 1981633767850818093910094, 258406311809937562215099482, 35994776359231593721760238102
Offset: 0

Views

Author

Paul D. Hanna, Jun 18 2015

Keywords

Examples

			E.g.f.: A(x) = 1 + 2*x + 18*x^2/2! + 302*x^3/3! + 7562*x^4/4! + 253542*x^5/5! +...
where A(3*x - 2*x*exp(x)) = 1/(3 - 2*exp(x)).

Crossrefs

Cf. A259063, A052894, A258923.

Programs

  • Mathematica
    CoefficientList[1/x*InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 21}], x],x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)
  • PARI
    {a(n) = local(A=1); A = (1/x)*serreverse(3*x - 2*x*exp(x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1); A = 1 + (1/x)*sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=1+x+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = (1/x) * Series_Reversion( 3*x - 2*x*exp(x) ).
(2) A(x) = 1 + (1/x) * Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
(3) A(x) = exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) = A259063(n+1) / (n+1). - Vaclav Kotesovec, Jun 19 2015
a(n) ~ (c/3)^(n+1) * n^(n-1) / (sqrt(c+1) * exp(n) * (c-1)^(2*n+1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015
a(n) = (1/(n+1)!) * Sum_{k=0..n} 2^k * (n+k)! * Stirling2(n,k). - Seiichi Manyama, Mar 06 2024

A259064 E.g.f.: Series_Reversion( 4*x - 3*x*exp(x) ).

Original entry on oeis.org

1, 6, 117, 3792, 172005, 10030248, 714843885, 60207412128, 5850995291397, 644410711219800, 79322681596610661, 10791841135527454896, 1608054016025580893445, 260445389091217967677992, 45557042043723212142506205, 8559094926999510089793332544, 1718950045690606262911636792677
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2015

Keywords

Examples

			E.g.f.: A(x) = x + 6*x^2/2! + 117*x^3/3! + 3792*x^4/4! + 172005*x^5/5! +...
where A(4*x - 3*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 6*x^2 + 117*x^3 + 3792*x^4 + 172005*x^5 + 10030248*x^6 +...
where F(x)/x = 1/4 + 3/(4-x)^2 + 3^2/(4-2*x)^3 + 3^3/(4-3*x)^4 + 3^4/(4-4*x)^5 +...

Links

Crossrefs

Cf. A053492, A259063, A259065, A259066.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[4*x - 3*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)
  • PARI
    {a(n) = local(A=x); A = serreverse(4*x - 3*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 3^m*(exp(x+x*O(x^n))-1)^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 3^m*(exp(x+x*O(x^n))-1)^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f.: x * Sum_{n>=0} 3^n / (4 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 3^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(4*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(4*exp(1)/3). - Vaclav Kotesovec, Jun 19 2015

A259065 E.g.f.: Series_Reversion( 5*x - 4*x*exp(x) ).

Original entry on oeis.org

1, 8, 204, 8656, 514100, 39254904, 3663341724, 404021632928, 51413304278916, 7414832746025800, 1195170934203398636, 212923233266007511152, 41545466670049713766356, 8811212141812890158250776, 2018230889016461893216938300, 496523506149784085749952075584, 130578628540561635331879674437156
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2015

Keywords

Examples

			E.g.f.: A(x) = x + 8*x^2/2! + 204*x^3/3! + 8656*x^4/4! + 514100*x^5/5! +...
where A(5*x - 4*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 8*x^2 + 204*x^3 + 8656*x^4 + 514100*x^5 + 39254904*x^6 +...
where F(x)/x = 1/5 + 4/(5-x)^2 + 4^2/(5-2*x)^3 + 4^3/(5-3*x)^4 + 4^4/(5-4*x)^5 +...

Links

Crossrefs

Cf. A053492, A259063, A259064, A259066.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[5*x - 4*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)
  • PARI
    {a(n) = local(A=x); A = serreverse(5*x - 4*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x+sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 4^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f.: x * Sum_{n>=0} 4^n / (5 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(5*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(5*exp(1)/4). - Vaclav Kotesovec, Jun 19 2015

A259066 E.g.f.: Series_Reversion( 6*x - 5*x*exp(x) ).

Original entry on oeis.org

1, 10, 315, 16520, 1212775, 114465780, 13204213435, 1800094703440, 283154358503295, 50478562633826300, 10057594831485171355, 2214859039031666012760, 534202513174577053611415, 140048168049127802257998820, 39652657811418543065286846075, 12058716801545122639605896216480
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2015

Keywords

Examples

			E.g.f.: A(x) = x + 10*x^2/2! + 315*x^3/3! + 16520*x^4/4! + 1212775*x^5/5! +...
where A(6*x - 5*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 10*x^2 + 315*x^3 + 16520*x^4 + 1212775*x^5 +...
where F(x)/x = 1/6 + 5/(6-x)^2 + 5^2/(6-2*x)^3 + 5^3/(6-3*x)^4 + 5^4/(6-4*x)^5 +...

Links

Crossrefs

Cf. A053492, A259063, A259064, A259065.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[6*x - 5*x*E^x, {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)
  • PARI
    {a(n) = local(A=x); A = serreverse(6*x - 5*x*exp(x +x*O(x^n) )); n!*polcoeff(A,n)}
for(n=1,20,print1(a(n),", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 5^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

O.g.f.: x * Sum_{n>=0} 5^n / (6 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 5^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(6*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(6*exp(1)/5). - Vaclav Kotesovec, Jun 19 2015

A259062 E.g.f.: Series_Reversion( -x + 2*x*exp(-x) ).

Original entry on oeis.org

1, 4, 42, 728, 17630, 548532, 20852370, 936655792, 48540537702, 2850727359500, 187107038833946, 13572973331551944, 1078343465147156910, 93119965280416893028, 8684514946963752624930, 869915871265946242868576, 93146889134541855185069942, 10617155946603647157142073916
Offset: 1

Views

Author

Paul D. Hanna, Jun 18 2015

Keywords

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 42*x^3/3! + 728*x^4/4! + 17630*x^5/5! +...
where A(-x + 2*x*exp(-x)) = x.

Crossrefs

Cf. A258872, A053492, A259063, A259064, A259065, A259066.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)
  • PARI
    {a(n) = local(A=x); A = serreverse(-x + 2*x*exp(-x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
  • PARI
    {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Formula

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).
a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^n * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - Vaclav Kotesovec, Jun 19 2015
Showing 1-5 of 5 results.

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