|
%I #8 Jul 08 2016 01:38:38
%S 1,1,2,6,20,141,260,1985,3887,124213,63013,2072737,4308063,72299423,
%T 19110234,1302089975,2789327371,192236392547,13000018109,
%U 3616336753079,7889633483913,138181586307115,37935218826111,2673839246416983,5905195803386319,209178924613512833,58019641542960071,4128415092111144721,9197893555411205235,164237433151443645047,11474995858629174895,3289139509736729288671,7376472520390786310211,1060236793997777905833443,74507380484700694744045,21473133267770763414163167
%N G.f. A(x) satisfies: A(x) = x + A(x)^2 - R(A(x)^2) + sqrt( A(x)^2 - R(A(x)^2) ), where R(A(x)) = x, and A(x) = Sum_{n>=1} a(n)*x^n/2^A274715(n).
%H Paul D. Hanna, <a href="/A274714/b274714.txt">Table of n, a(n) for n = 1..1040</a>
%F Given g.f. A(x) = Sum_{n>=1} a(n)*x^n / 2^A274715(n), then A(x) satisfies:
%F (1) A(x) = x + G( A(x)^4 ) + sqrt( G( A(x)^4 ) ), where G( A(x)^2 ) = A(x) - x, and G(x) is the g.f. described by A274717.
%F (2) A(x) = limit (F_{n} - x)^(1/2^n), where F_{n+1} = x + (F_{n+1} - F_{n})^2, starting with F_1 = A(x).
%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 141/2*x^6 + 260*x^7 + 1985/2*x^8 + 3887*x^9 + 124213/8*x^10 + 63013*x^11 + 2072737/8*x^12 + 4308063/4*x^13 + 72299423/16*x^14 + 19110234*x^15 + 1302089975/16*x^16 + 2789327371/8*x^17 + 192236392547/128*x^18 + 13000018109/2*x^19 + 3616336753079/128*x^20 + 7889633483913/64*x^21 + 138181586307115/256*x^22 + 37935218826111/16*x^23 + 2673839246416983/256*x^24 + 5905195803386319/128*x^25 + 209178924613512833/1024*x^26 + 58019641542960071/64*x^27 + 4128415092111144721/1024*x^28 + 9197893555411205235/512*x^29 + 164237433151443645047/2048*x^30 + 11474995858629174895/32*x^31 + 3289139509736729288671/2048*x^32 +...+ a(n)*x^n/2^A274715(n) +...
%e such that A(x) = x + A(x)^2 - R(A(x)^2) + sqrt(A(x)^2 - R(A(x)^2)), where R(A(x)) = x.
%e RELATED SERIES.
%e Let G(x) satisfy: (G(x) - G(x^2))^2 = G(x^2), where
%e G(x) = x + x^2 + 1/2*x^3 + x^4 + 1/8*x^5 + 1/2*x^6 + 7/16*x^7 + x^8 - 21/128*x^9 + 1/8*x^10 + 71/256*x^11 + 1/2*x^12 + 5/1024*x^13 + 7/16*x^14 + 1095/2048*x^15 + x^16 +...+ A274717(n)*x^n/2^A274716(n) +...
%e then the series reversion of A(x) equals x - G(x^2) and begins:
%e x - G(x^2) = x - x^2 - x^4 - 1/2*x^6 - x^8 - 1/8*x^10 - 1/2*x^12 - 7/16*x^14 - x^16 + 21/128*x^18 - 1/8*x^20 - 71/256*x^22 - 1/2*x^24 - 5/1024*x^26 - 7/16*x^28 - 1095/2048*x^30 - x^32 +...+ -A274717(n)*x^(2*n)/2^A274716(n) +...
%e Also, A(x) = x + G(A(x)^4) + sqrt( G(A(x)^4) ).
%e AS THE LIMIT OF AN ITERATED PROCESS.
%e If we start with F1 = A(x), and define functions F_{n} that satisfy:
%e (1) F2 = x + (F2 - F1)^2
%e (2) F3 = x + (F3 - F2)^2
%e (3) F4 = x + (F4 - F3)^2
%e and continue in this way,
%e F_{n+1} = x + (F_{n+1} - F_{n})^2,
%e then
%e limit (F_{n} - x)^(1/2^n) = A(x).
%e The initial functions described above begin:
%e (1) F2 = x + x^4 + 4*x^5 + 14*x^6 + 52*x^7 + 202*x^8 + 802*x^9 + 3234*x^10 + 13220*x^11 + 109357/2*x^12 + 456813/2*x^13 + 962236*x^14 + 4083620*x^15 + 17442222*x^16 +...
%e (2) F3 = x + x^8 + 8*x^9 + 44*x^10 + 216*x^11 + 1014*x^12 + 4652*x^13 + 21064*x^14 + 94664*x^15 + 423658*x^16 + 1891885*x^17 + 8440218*x^18 + 37646850*x^19 +...
%e (3) F4 = x + x^16 + 16*x^17 + 152*x^18 + 1136*x^19 + 7420*x^20 + 44536*x^21 + 252448*x^22 + 1373776*x^23 + 7253430*x^24 + 37426586*x^25 + 189685596*x^26 +...
%e where the limit of the roots approach g.f. A(x):
%e (1) (F2 - x)^(1/4) = x + x^2 + 2*x^3 + 6*x^4 + 81/4*x^5 + 287/4*x^6 + 265*x^7 + 2025/2*x^8 +...
%e (2) (F3 - x)^(1/8) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 141/2*x^6 + 260*x^7 + 1985/2*x^8 + 31097/8*x^9 +...
%e (3) (F4 - x)^(1/16) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 141/2*x^6 + 260*x^7 + 1985/2*x^8 + 3887*x^9 +...
%e etc.
%o (PARI) {a(n) = my(A=x+x^2,R=x); for(i=1,n,
%o R = serreverse(A + x^2*O(x^n));
%o A = x + A^2 - subst(R,x,A^2) + sqrt(A^2 - subst(R,x,A^2)) );
%o numerator(polcoeff(A,n))}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A274717, A274716, A274715.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jul 08 2016
| 