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 1-3 of 3 results.

A302583 a(n) = ((n + 1)^n - (n - 1)^n)/2.

Original entry on oeis.org

0, 1, 4, 28, 272, 3376, 51012, 908608, 18640960, 432891136, 11225320100, 321504185344, 10079828372880, 343360783937536, 12627774819845668, 498676704524517376, 21046391759976988928, 945381827279671853056, 45032132922921758270916, 2267322327322331161821184
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 10 2018

Keywords

Links

Crossrefs

Cf. A000169, A065440, A007778, A062024, A115416, A274278, A293022, A302584, A302585, A302586, A302587.

Programs

  • Mathematica
    Table[((n + 1)^n - (n - 1)^n)/2, {n, 0, 19}]
nmax = 19; CoefficientList[Series[(x^2 - LambertW[-x]^2)/(2 x LambertW[-x] (1 + LambertW[-x])), {x, 0, nmax}], x] Range[0, nmax]!
Table[n! SeriesCoefficient[Exp[n x] Sinh[x], {x, 0, n}], {n, 0, 19}]

Formula

E.g.f.: (x^2 - LambertW(-x)^2)/(2*x*LambertW(-x)*(1 + LambertW(-x))).
a(n) = n! * [x^n] exp(n*x)*sinh(x).

A274279 Expansion of e.g.f.: tanh(x*W(x)), where W(x) = LambertW(-x)/(-x).

Original entry on oeis.org

1, 2, 7, 40, 341, 3936, 57107, 992384, 20025385, 459466240, 11804134079, 335571265536, 10456512176189, 354362575314944, 12975301760361163, 510462668072058880, 21472710312090391889, 961728814178702327808, 45692671937666739799799, 2295278998002033651875840, 121545436687537993689631525, 6767130413049423041105231872, 395177438856180565803457658627, 24152146710231984411570685870080
Offset: 1

Views

Author

Paul D. Hanna, Jun 19 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 7*x^3/3! + 40*x^4/4! + 341*x^5/5! + 3936*x^6/6! + 57107*x^7/7! + 992384*x^8/8! + 20025385*x^9/9! + 459466240*x^10/10! + 11804134079*x^11/11! + 335571265536*x^12/12! +...
such that A(x) = tanh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x)^2 - 1)/(W(x)^2 + 1), where
W(x)^2 = 1 + 2*x + 8*x^2/2! + 50*x^3/3! + 432*x^4/4! + 4802*x^5/5! + 65536*x^6/6! + 1062882*x^7/7! + 20000000*x^8/8! +...+ 2*(n+2)^(n-1)*x^n/n! +...

Links

Crossrefs

Cf. A000272, A195136, A238085, A264902, A274278, A277468, A277480.

Programs

  • Mathematica
    Rest[CoefficientList[Series[(LambertW[-x]^2 - x^2)/(LambertW[-x]^2 + x^2), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 23 2016 *)
Rest[With[{nmax=30}, CoefficientList[Series[Tanh[-LambertW[-x]], {x,0,nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Feb 19 2018 *)
  • PARI
    {a(n) = my(W=sum(m=0, n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff(tanh(x*W), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    {a(n) = my(W = sum(m=0, n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff( (W^2 - 1)/(W^2 + 1), n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    x='x+O('x^30); Vec(serlaplace(tanh(-lambertw(-x)))) \\ G. C. Greubel, Feb 19 2018

Formula

E.g.f.: (W(x)^2 - 1)/(W(x)^2 + 1), where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) ~ 4*exp(2) * n^(n-1) / (1+exp(2))^2. - Vaclav Kotesovec, Jun 23 2016
a(n) = Sum_{k=0..n-1} (-1)^k * A264902(n,k). - Alois P. Heinz, Aug 08 2022

A195136 a(n) = ((n+1)^(n-1) + (n-1)^(n-1))/2 for n>=1.

Original entry on oeis.org

1, 2, 10, 76, 776, 9966, 154400, 2803256, 58388608, 1372684090, 35958682112, 1038736032324, 32805006411776, 1124535087475814, 41584800431742976, 1650158470945337584, 69943137585151901696, 3153813559835569475058, 150745204037648268787712, 7613458147995669857352380, 405143549343202022103973888, 22657085569540734204315357022, 1328470689420203636727039918080, 81494507575933974604289943213096, 5220210773193749540624447754469376, 348542314841685116176787263033063466, 24216786265392720787141148530274467840, 1748280517106781152846793195054531026356, 130956723831431687431286364126682302906368, 10164786953127554557192799138093559445158870
Offset: 1

Views

Author

Paul D. Hanna, Sep 09 2011

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 10*x^3/3! + 76*x^4/4! + 776*x^5/5! + 9966*x^6/6! + 154400*x^7/7! + 2803256*x^8/8! + 58388608*x^9/9! + 1372684090*x^10/10! +...
such that A(x) = sinh(x*W(x))
where W(x) = LambertW(-x)/(-x) begins
W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
and satisfies W(x) = exp(x*W(x)).
Also, A(x) = (W(x) - 1/W(x))/2 where
1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +...

Crossrefs

Cf. A000272, A274278, A274279.

Programs

  • Mathematica
    Join[{1},Table[((n+1)^(n-1)+(n-1)^(n-1))/2,{n,2,30}]] (* Harvey P. Dale, Feb 06 2023 *)
  • PARI
    {a(n)=((n+1)^(n-1) + (n-1)^(n-1))/2}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=sum(k=0,(n-1)\2,binomial(n-1,2*k)*n^(n-2*k-1))}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=local(W=sum(m=0,n,(m+1)^(m-1)*x^m/m!)+x*O(x^n));n!*polcoeff(sinh(x*W),n)}
for(n=1,30,print1(a(n),", "))

Formula

E.g.f.: sinh(x*W(x)) = (W(x) - 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k) * n^(n-2*k-1).

Extensions

Entry revised by Paul D. Hanna, Jun 19 2016
Corrected and extended by Harvey P. Dale, Feb 06 2023
Showing 1-3 of 3 results.

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