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.

A274393 E.g.f. A(x) satisfies: A( A( x^3*exp(-3*x) )^(1/3) ) = x.

Original entry on oeis.org

1, 2, 9, 56, 545, 6696, 100009, 1756112, 35480673, 811332080, 20696592521, 583009540488, 17972297981521, 601961695890296, 21765379980020265, 844991974575946016, 35056808550027808961, 1547900555791042958688, 72474037424646843142153, 3586609339433026549298840, 187062738581835989450074161, 10255505482370456224398408872, 589611389520200188085133153449
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 56*x^4/4! + 545*x^5/5! + 6696*x^6/6! + 100009*x^7/7! + 1756112*x^8/8! + 35480673*x^9/9! + 811332080*x^10/10! + 20696592521*x^11/11! + 583009540488*x^12/12! + 17972297981521*x^13/13! + 601961695890296*x^14/14! + 21765379980020265*x^15/15! +...
such that A( A( x^3*exp(-3*x) )^(1/3) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^3*exp(-3*x) )^(1/3) = x - 2*x^2/2! + 3*x^3/3! + 4*x^4/4! - 155*x^5/5! + 1914*x^6/6! - 15953*x^7/7! + 33592*x^8/8! + 2425257*x^9/9! - 71955530*x^10/10! + 1307665051*x^11/11! - 13707439692*x^12/12! - 125013414227*x^13/13! + 11742108426034*x^14/14! - 370418656051065*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^3 * exp(-3*A(x)) = x^3 - 2*x^6/2! + 3*x^9/3! + 4*x^12/4! - 155*x^15/5! + 1914*x^18/6! - 15953*x^21/7! + 33592*x^24/8! +...
where A( A(x)^3 * exp(-3*A(x)) ) = x^3.
Note the following series is also in powers of x^3:
A(-A(x)^3 * exp(-3*A(x)) ) = -x^3 + 4*x^6/2! - 24*x^9/3! + 208*x^12/4! - 2400*x^15/5! + 36432*x^18/6! - 700672*x^21/7! + 16221088*x^24/8! - 434076480*x^27/9! + 13091390560*x^30/10! - 438602465664*x^33/11! + 16177344184080*x^36/12! - 652301794869088*x^39/13! + 28571154198355888*x^42/14! +...

Links

Crossrefs

Cf. A274275, A274394, A274395.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^3*exp(-3*x +x*O(x^n)))^(1/3) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( A( x^3*exp(3*x) )^(1/3) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( A( x^3*exp(-3*x) )^(1/3) ).
(3) A( A(x)^3 * exp(-3*A(x)) ) = x^3.
a(n) ~ c * n! * d^n / n^(3/2), where d = 2.67119188675..., c = 0.3883596303... . - Vaclav Kotesovec, Aug 12 2021

A274394 E.g.f. A(x) satisfies: A( A( x^4*exp(-4*x) )^(1/4) ) = x.

Original entry on oeis.org

1, 2, 9, 64, 595, 7416, 111979, 1989632, 40695561, 941667040, 24323649361, 693818707968, 21661372820971, 734712173277824, 26902827107293635, 1057724890214957056, 44442356900221356241, 1987370544970750468608, 94240073170115929379161, 4723448516579307027169280, 249510355552473169494452931, 13854414947224528743034304512, 806733172355775780726416256859
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2016

Keywords

Examples

			E.g.f.: A(x) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 595*x^5/5! + 7416*x^6/6! + 111979*x^7/7! + 1989632*x^8/8! + 40695561*x^9/9! + 941667040*x^10/10! + 24323649361*x^11/11! + 693818707968*x^12/12! + 21661372820971*x^13/13! + 734712173277824*x^14/14! + 26902827107293635*x^15/15! + 1057724890214957056*x^16/16! +...
such that A( A( x^4*exp(-4*x) )^(1/4) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^4*exp(-4*x) )^(1/4) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 35*x^5/5! - 906*x^6/6! + 15757*x^7/7! - 210008*x^8/8! + 2464569*x^9/9! - 32810410*x^10/10! + 671239811*x^11/11! - 18224632812*x^12/12! + 496597765963*x^13/13! - 12681217528994*x^14/14! + 320976165059565*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^4 * exp(-4*A(x)) = x^4 - 2*x^8/2! + 3*x^12/3! - 4*x^16/4! + 35*x^20/5! - 906*x^24/6! + 15757*x^28/7! - 210008*x^32/8! + 2464569*x^36/9! - 32810410*x^40/10! +...
where A( A(x)^4 * exp(-4*A(x)) ) = x^4.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( A(x^4*exp(4*x))^(1/4) ) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47232*x^6/6! + 942592*x^7/7! + 22171648*x^8/8! +...+ A216857(n)*x^n/n! +...
which equals -LambertW(-x*exp(x)).

Links

Crossrefs

Cf. A274275, A274393, A274395.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^4*exp(-4*x +x*O(x^n)))^(1/4) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( A( x^4*exp(4*x) )^(1/4) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( A( x^4*exp(-4*x) )^(1/4) ).
(3) A( A(x)^4 * exp(-4*A(x)) ) = x^4.
(4) A(-A(x)^4 * exp(-4*A(x)) ) = -LambertW(x^4*exp(-x^4)).

A274395 E.g.f. A(x) satisfies: A( A( x^5*exp(-5*x) )^(1/5) ) = x.

Original entry on oeis.org

1, 2, 9, 64, 625, 7632, 115633, 2060864, 42272577, 981100000, 25420901209, 727392785472, 22781551770289, 775174385740496, 28475611427390625, 1123174379270470528, 47345176946662808833, 2124056646149570472384, 101049649535116764217513, 5081280208216339430000000, 269289663191356712678537841, 15001629187601225176466619952, 876397229390129697388339920049
Offset: 1

Views

Author

Paul D. Hanna, Jun 21 2016

Keywords

Examples

			E.g.f.: A(x) =  x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7632*x^6/6! + 115633*x^7/7! + 2060864*x^8/8! + 42272577*x^9/9! + 981100000*x^10/10! + 25420901209*x^11/11! + 727392785472*x^12/12! + 22781551770289*x^13/13! + 775174385740496*x^14/14! + 28475611427390625*x^15/15! +...
such that A( A( x^5*exp(-5*x) )^(1/5) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
A( x^5*exp(-5*x) )^(1/5) = x - 2*x^2/2! + 3*x^3/3! - 4*x^4/4! + 5*x^5/5! + 138*x^6/6! - 6041*x^7/7! + 145144*x^8/8! - 2612727*x^9/9! + 39191030*x^10/10! - 508540021*x^11/11! + 5048676852*x^12/12! + 13708341517*x^13/13! - 3528271498766*x^14/14! + 168238690139535*x^15/15! +...
Compare the above series reversion to the following series:
A(x)^5 * exp(-5*A(x)) = x^5 - 2*x^10/2! + 3*x^15/3! - 4*x^20/4! + 5*x^25/5! + 138*x^30/6! - 6041*x^35/7! + 145144*x^40/8! - 2612727*x^45/9! +...
where A( A(x)^5 * exp(-5*A(x)) ) = x^5.
Note the following series is also in powers of x^5:
A(-A(x)^5 * exp(-5*A(x)) ) = -x^5 + 4*x^10/2! - 24*x^15/3! + 224*x^20/4! - 2880*x^25/5! + 46944*x^30/6! - 926464*x^35/7! + 21582976*x^40/8! - 581587200*x^45/9! + 17791870400*x^50/10! - 608466076416*x^55/11! + 22988808466560*x^60/12! - 950707246757632*x^65/13! + 42712071655886752*x^70/14! - 2071447871355102720*x^75/15! + 107861157493089761024*x^80/16! +...

Links

Crossrefs

Cf. A274275, A274393, A274394.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^5*exp(-5*x +x*O(x^n)))^(1/5) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / n! satisfies:
(1) A( A( x^5*exp(5*x) )^(1/5) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( A( x^5*exp(-5*x) )^(1/5) ).
(3) A( A(x)^5 * exp(-5*A(x)) ) = x^5.

A274276 E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x, where A(x) = Sum_{n>=1} a(n)*x^n/(n-1)!.

Original entry on oeis.org

1, 1, 2, 10, 80, 776, 8992, 130768, 2252672, 43823872, 957193856, 23369928704, 629680631296, 18514472015872, 590350181439488, 20311856724176896, 749913022501879808, 29561045244530032640, 1239353203580700000256, 55077035791625925492736, 2586090541400666789543936, 127922890235433583945056256, 6649362432158408977810522112, 362360171399316029979428126720, 20658795751396952768159379619840
Offset: 1

Views

Author

Paul D. Hanna, Jun 17 2016

Keywords

Examples

			E.g.f.: A(x) = x + x^2 + 2*x^3/2! + 10*x^4/3! + 80*x^5/4! + 776*x^6/5! + 8992*x^7/6! + 130768*x^8/7! + 2252672*x^9/8! + 43823872*x^10/9! + 957193856*x^11/10! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.

Links

Crossrefs

Cf. A274275.

Programs

  • PARI
    {a(n) = my(A=x); for(i=1, n, A = serreverse( sqrt( subst(A, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); (n-1)!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

a(n) = A274275(n)/n.
E.g.f. A(x) = Sum_{n>=1} a(n) * x^n / (n-1)! satisfies:
(1) A( sqrt( A(x^2*exp(2*x)) ) ) = -LambertW(-x*exp(x)).
(2) A(x) = Series_Reversion( sqrt( A(x^2*exp(-x)) ) ).
(3) A( A(x)^2 * exp(-2*A(x)) ) = x^2.
(4) A(-A(x)^2 * exp(-2*A(x)) ) = -LambertW(x^2*exp(-x^2)).
a(n)/n! ~ c * d^n / n^(5/2), where d = 2.52462188117..., c = 0.36965356... . - Vaclav Kotesovec, Jun 23 2016

A274277 E.g.f. A(x) satisfies: A( A(x)^2 ) = x^2 * exp(-2*x).

Original entry on oeis.org

1, -2, 6, -40, 320, -2976, 35392, -538112, 9931392, -211790080, 5059784576, -132643057152, 3761875287040, -114501941915648, 3725395402721280, -129324055589257216, 4786638435256696832, -188785468724361560064, 7922155381738193944576, -352740315643746941665280, 16603695476218208847691776, -822951583413551750366298112, 42792449844854211313594597376, -2327246576567999111735900897280, 132052357036729088907927420928000
Offset: 1

Views

Author

Paul D. Hanna, Jun 18 2016

Keywords

Examples

			E.g.f.: A(x) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 320*x^5/5! - 2976*x^6/6! + 35392*x^7/7! - 538112*x^8/8! + 9931392*x^9/9! - 211790080*x^10/10! + 5059784576*x^11/11! - 132643057152*x^12/12! + 3761875287040*x^13/13! - 114501941915648*x^14/14! + 3725395402721280*x^15/15! - 129324055589257216*x^16/16! +...
where A( A(x)^2 ) = x^2 * exp(-2*x).
RELATED SERIES.
Let B(x) be the series reversion of the e.g.f. A(x), which begins
B(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 400*x^5/5! + 4656*x^6/6! + 62944*x^7/7! + 1046144*x^8/8! + 20274048*x^9/9! + 438238720*x^10/10! + 10529132416*x^11/11! + 280439144448*x^12/12! + 8185848206848*x^13/13! + 259202608222208*x^14/14! +...+ A274275(n)*x^n/n! +...
then A(x) = sqrt( B( x^2*exp(-2*x) ) )
and A(x^2) = B(x)^2 * exp(-2*B(x)).
A(x)^2 = 2*x^2/2! - 12*x^3/3! + 72*x^4/4! - 640*x^5/5! + 6960*x^6/6! - 85344*x^7/7! + 1226624*x^8/8! - 21007872*x^9/9! + 422254080*x^10/10! - 9724042240*x^11/11! + 250998494208*x^12/12! +...
where A(x)^2 = B( x^2*exp(-2*x) ) such that B(A(x)) = x.

Crossrefs

Cf. A274275.

Programs

  • PARI
    /* From A(x) = sqrt( B( x^2*exp(-2*x) ) ) where A(B(x)) = x */
{a(n) = my(A=x,B=x); for(i=1, n, B = serreverse(A +x*O(x^n)); A = sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ); n!*polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
  • PARI
    /* As the series reversion of the e.g.f. of A274275 */
{a(n) = my(B=x); for(i=1, n, B = serreverse( sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); n!*polcoeff(serreverse(B), n)}
for(n=1, 30, print1(a(n), ", "))

Formula

E.g.f. equals the series reversion of the e.g.f. of A274275.
Given e.g.f. A(x), let B(x) satisfy A(B(x)) = x, then
(1) A(x) = sqrt( B( x^2*exp(-2*x) ) ).
(2) A(x^2) = B(x)^2 * exp(-2*B(x)).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.