%I #36 Aug 18 2016 13:28:36

%S 1,1,2,4,9,20,46,104,238,540,1228,2780,6289,14180,31924,71688,160694,

%T 359452,802642,1788988,3980916,8844064,19618506,43455324,96121164,

%U 212331796,468445180,1032216460,2271818652,4994434788,10968013396,24061103888,52730956193,115449870424,252530306764,551873275488,1204991320660,2628810554176,5730295148952,12480957518212,27163290056278

%N G.f. A(x) satisfies: 1 = ...(((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2) -...- x^n)^(1/2) -..., an infinite series of nested square roots.

%C Odd terms occur at positions k*2^(k-1) for k>=0.

%C Limit a(n+1)/a(n) = 2, and A(x) diverges at x=1/2.

%C A(-1/2) = 1.0891636602638152861240865158090054430536947422594419370337760...

%C A(2/5) = 4.27983467184471084235872646732512184377478311914374590...

%C A(1/3) = 2.15485192359458408375371476779655861137906655796801630...

%C A(x) = 2 at x = 0.32026273178798900824351068844199852911740930864617900985902...

%H Paul D. Hanna, <a href="/A274965/b274965.txt">Table of n, a(n) for n = 0..1030</a>

%F G.f.: A(x) = G(x,1), where G(x,y) = x*y + G(x,x*y)^2 is the g.f. of A275670.

%F G.f.: A(x) = F(x)^2 + x, where F(x) is the g.f. of A275691.

%e G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 104*x^7 + 238*x^8 + 540*x^9 + 1228*x^10 +...

%e Illustration of the definition.

%e R1 = (A(x) - x)^(1/2);

%e R2 = ((A(x) - x)^(1/2) - x^2)^(1/2);

%e R3 = (((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2);

%e R4 = ((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2);

%e R5 = (((((A(x) - x)^(1/2) - x^2)^(1/2) - x^3)^(1/2) - x^4)^(1/2) - x^5)^(1/2); ...

%e where the above series begin:

%e R1 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 36*x^7 + 78*x^8 + 168*x^9 + 364*x^10 + 786*x^11 + 1700*x^12 +...

%e R2 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + 16*x^7 + 33*x^8 + 68*x^9 + 142*x^10 + 296*x^11 + 620*x^12 + 1296*x^13 +...

%e R3 = 1 + x^4 + 2*x^5 + 4*x^6 + 8*x^7 + 16*x^8 + 32*x^9 + 65*x^10 + 132*x^11 + 270*x^12 + 552*x^13 + 1132*x^14 +...

%e R4 = 1 + x^5 + 2*x^6 + 4*x^7 + 8*x^8 + 16*x^9 + 32*x^10 + 64*x^11 + 129*x^12 + 260*x^13 + 526*x^14 + 1064*x^15 +...

%e R5 = 1 + x^6 + 2*x^7 + 4*x^8 + 8*x^9 + 16*x^10 + 32*x^11 + 64*x^12 + 128*x^13 + 257*x^14 + 516*x^15 + 1038*x^16 +...

%e etc., so that 1 is obtained as a limit.

%e GENERATING METHOD.

%e The g.f. of this sequence can be obtained as a limit, as n grows, of the following process: start with 1 + x^n, then square the result and add x^(n-1), then square the result and add x^(n-2), then continue in this way until you reach x^1; this process is illustrated at n=6 as follows:

%e S6 = 1 + x^6,

%e S5 = S6^2 + x^5 = 1 + x^5 + 2*x^6 + x^12,

%e S4 = S5^2 + x^4 = 1 + x^4 + 2*x^5 + 4*x^6 + x^10 + 4*x^11 + 6*x^12 + 2*x^17 +...,

%e S3 = S4^2 + x^3 = 1 + x^3 + 2*x^4 + 4*x^5 + 8*x^6 + x^8 + 4*x^9 + 14*x^10 +...,

%e S2 = S3^2 + x^2 = 1 + x^2 + 2*x^3 + 4*x^4 + 8*x^5 + 17*x^6 + 4*x^7 + 14*x^8 + 40*x^9 + 76*x^10 +...,

%e S1 = S2^2 + x = 1 + x + 2*x^2 + 4*x^3 + 9*x^4 + 20*x^5 + 46*x^6 + 40*x^7 + 110*x^8 + 220*x^9 + 396*x^10 +...,

%e which matches the g.f. A(x) up to x^6.

%e RELATED SERIES.

%e Note that the bisections are self-convolutions of integer sequences:

%e sqrt( (A(x) + A(-x))/2 ) = 1 + x^2 + 4*x^4 + 19*x^6 + 92*x^8 + 446*x^10 + 2150*x^12 + 10280*x^14 + 48761*x^16 + 229558*x^18 + 1073278*x^20 + 4986624*x^22 + 23037102*x^24 + 105877968*x^26 + 484337300*x^28 +...+ A275751(n)*x^(2*n) +...

%e sqrt( x*(A(x) - A(-x))/2 ) = x + 2*x^3 + 8*x^5 + 36*x^7 + 166*x^9 + 770*x^11 + 3574*x^13 + 16560*x^15 + 76516*x^17 + 352498*x^19 + 1619014*x^21 + 7414134*x^23 + 33855996*x^25 + 154181234*x^27 + 700333366*x^29 +...+ A275752(n)*x^(2*n+1) +...

%o (PARI) {a(n) = my(A=1 +x*O(x^n)); for(k=0,n, A = A^2 + x^(n+1-k)); polcoeff(A,n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A275691, A275751, A275752.

%Y Row sums of triangle A275670.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 16 2016