A274275
E.g.f. A(x) satisfies: A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
Original entry on oeis.org
1, 2, 6, 40, 400, 4656, 62944, 1046144, 20274048, 438238720, 10529132416, 280439144448, 8185848206848, 259202608222208, 8855252721592320, 324989707586830336, 12748521382531956736, 532098814401540587520, 23547710868033300004864, 1101540715832518509854720, 54307901369414002580422656, 2814303585179538846791237632, 152935335939643406489642008576, 8696644113583584719506275041280, 516469893784923819203984490496000
Offset: 1
E.g.f.: A(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! + 8855252721592320*x^15/15! + 324989707586830336*x^16/16! +...
such that A( sqrt( A(x^2*exp(-2*x)) ) ) = x.
RELATED SERIES.
The series reversion of the e.g.f. A(x) equals the series defined by:
sqrt( A(x^2*exp(-2*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! +...+ A274277(n)*x^n/n! +...
Compare the above series reversion to the following series:
A(x)^2 * exp(-2*A(x)) = x^2 - 2*x^4/2! + 6*x^6/3! - 40*x^8/4! + 320*x^10/5! - 2976*x^12/6! + 35392*x^14/7! - 538112*x^16/8! + 9931392*x^18/9! +...
where A( A(x)^2 * exp(-2*A(x)) ) = x^2.
The e.g.f. A(x) is related to the LambertW function by the composition:
A( sqrt( A(x^2*exp(2*x)) ) ) = 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)).
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{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!*polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))
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
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)).
-
{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), ", "))
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
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! +...
-
{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), ", "))
Showing 1-3 of 3 results.