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 A295760 G.f. A(x) satisfies: A(x - A(x^2)) = x + A(x^2). 3

%I #14 Nov 27 2017 13:43:45

%S 1,2,4,14,52,204,840,3586,15708,70196,318792,1467068,6826360,32062184,

%T 151805328,723785606,3472055348,16745865716,81154862712,394993248572,

%U 1929956966024,9463011127592,46547604953424,229631732978956,1135868927282840,5632406184568280,27992869133724208,139417039713595032,695719759056964304,3478120152935676720,17417832347321830432

%N G.f. A(x) satisfies: A(x - A(x^2)) = x + A(x^2).

%H Paul D. Hanna, <a href="/A295760/b295760.txt">Table of n, a(n) for n = 1..1030</a>

%F G.f. A(x) satisfies:

%F (1) A(x) = x + 2 * A( (x + A(x))^2/4 ).

%F (2) A(x) = -x + 2 * Series_Reversion( x - A(x^2) ).

%F (3) x = A( -x + 2 * Series_Reversion( x + A(x^2) ) ).

%F (4) x*A(x) = (x + A(x))^2/4 - A( (x + A(x))^2/4 )^2.

%F (5) A(x - A(x^2)) = x + A(x^2).

%F (6) A(-A(-x)) = x.

%e G.f.: A(x) = x + 2*x^2 + 4*x^3 + 14*x^4 + 52*x^5 + 204*x^6 + 840*x^7 + 3586*x^8 + 15708*x^9 + 70196*x^10 + 318792*x^11 + 1467068*x^12 + 6826360*x^13 + 32062184*x^14 + 151805328*x^15 + 723785606*x^16 + 3472055348*x^17 + 16745865716*x^18 + 81154862712*x^19 + 394993248572*x^20 +...

%e such that A(x - A(x^2)) = x + A(x^2).

%e RELATED SERIES.

%e A(x - A(x^2)) = x + x^2 + 2*x^4 + 4*x^6 + 14*x^8 + 52*x^10 + 204*x^12 + 840*x^14 + 3586*x^16 + 15708*x^18 + 70196*x^20 +...

%e which equals x + A(x^2).

%e Series_Reversion( x - A(x^2) ) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35098*x^10 +...

%e which equals (A(x) + x)/2.

%e A( (x + A(x))^2/4 ) = x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35098*x^10 +...

%e which equals (A(x) - x)/2.

%e (x + A(x))^2/4 = x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 70*x^6 + 284*x^7 + 1197*x^8 + 5198*x^9 + 23078*x^10 +...

%e which equals x*A(x) + A( (x + A(x))^2/4 )^2.

%e Let B(B(x)) = A(x) then B(x) is an integer series (verified up to 400 terms):

%e B(x) = x + x^2 + x^3 + 4*x^4 + 10*x^5 + 33*x^6 + 105*x^7 + 360*x^8 + 1244*x^9 + 4350*x^10 + 15488*x^11 + 55514*x^12 + 201220*x^13 + 735409*x^14 + 2707973*x^15 + 10036908*x^16 + 37413444*x^17 + 140192022*x^18 + 527728468*x^19 + 1994613008*x^20 + 7566519020*x^21 + 28803657194*x^22 + 110000675444*x^23 + 421172979138*x^24 + 1616154840122*x^25 + 6220675694876*x^26 + 24028744940126*x^27 + 92796758654138*x^28 + 357109198506472*x^29 + 1389208125591993*x^30 + 5552344056227841*x^31 + 21323110914365336*x^32 + 70454013218649400*x^33 + 298345385254918858*x^34 + 2355991303858543108*x^35 + 6997638978589417444*x^36 - 100308304079135213248*x^37 - 153429582192527911554*x^38 + 14890888637965051478428*x^39 + 29224269440712871606248*x^40 +...

%e where B(x - A(x^2)) is an odd function that begins

%e B(x - A(x^2)) = x - x^3 - 7*x^5 - 37*x^7 - 189*x^9 - 583*x^11 + 1255*x^13 + 28711*x^15 + 218937*x^17 + 890823*x^19 + 760249*x^21 - 69490111*x^23 + 649102437*x^25 - 54674874881*x^27 + 3035063408777*x^29 - 209576545439765*x^31 + 16859178743641679*x^33 - 1563274578274583407*x^35 + 165493803731758762623*x^37 - 19828488694059640880745*x^39 +...

%e also B(-B(-x) = x.

%o (PARI) {a(n) = my(A=x); for(i=1,n, A = -x + 2*serreverse(x - subst(A,x,x^2) +x^2*O(x^n)) ); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A141200.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 26 2017

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Last modified September 21 02:47 EDT 2023. Contains 365486 sequences. (Running on oeis4.)