A192738 - OEIS

%I #13 Nov 22 2017 19:31:25

%S 1,1,2,6,19,64,225,817,3037,11503,44237,172274,678012,2692519,

%T 10775673,43416923,175973241,716988597,2935005104,12065005233,

%U 49784012457,206130676553,856155851503,3566191166505,14893485428089,62350269656535,261608048386819,1099926940133938,4633543246538166,19554304381037925,82660879651195338

%N G.f. satisfies: A(x) = 1/(1 - x*A(x)/(1 - x^2*A(x)/(1 - x^4*A(x)/(1 - x^8*A(x)/(1 - ...))))), a recursive continued fraction.

%H Vaclav Kotesovec, <a href="/A192738/b192738.txt">Table of n, a(n) for n = 0..700</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 4.441030346767249462955690731977198943168711119968067173405434... and c = 0.53428706465230936021267408350544799998698165793457... - _Vaclav Kotesovec_, Nov 18 2017

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 225*x^6 + 817*x^7 + 3037*x^8 + 11503*x^9 + 44237*x^10 + 172274*x^11 + 678012*x^12 + 2692519*x^13 + 10775673*x^14 + 43416923*x^15 + 175973241*x^16 + 716988597*x^17 + 2935005104*x^18 + 12065005233*x^19 + 49784012457*x^20 +...

%e The g.f. satisfies:

%e A(x) = 1 + x*A(x) + (x^2 + x^3)*A(x)^2 + (x^3 + 2*x^4 + x^5 + x^7)*A(x)^3 + (x^4 + 3*x^5 + 3*x^6 + x^7 + 2*x^8 + 2*x^9 + x^11 + x^15)*A(x)^4 + (x^5 + 4*x^6 + 6*x^7 + 4*x^8 + 4*x^9 + 6*x^10 + 3*x^11 + 2*x^12 + 3*x^13 + x^15 + 2*x^16 + 2*x^17 + 2*x^19 + x^23 + x^31)*A(x)^5 + (x^6 + 5*x^7 + 10*x^8 + 10*x^9 + 9*x^10 + 13*x^11 + 12*x^12 + 7*x^13 + 9*x^14 + 6*x^15 + 2*x^16 + 7*x^17 + 6*x^18 + 4*x^19 + 4*x^20 + 6*x^21 + 3*x^23 + 2*x^24 + 2*x^25 + 3*x^27 + x^31 + 2*x^32 + 2*x^33 + 2*x^35 + 2*x^39 + x^47 + x^63)*A(x)^6 + (x^7 + 6*x^8 + 15*x^9 + 20*x^10 + 20*x^11 + 26*x^12 + 31*x^13 + 24*x^14 + 23*x^15 + 24*x^16 + 13*x^17 + 16*x^18 + 22*x^19 + 14*x^20 + 15*x^21 + 18*x^22 + 13*x^23 + 6*x^24 + 15*x^25 + 6*x^26 + 7*x^27 + 6*x^28 + 9*x^29 + 6*x^31 + 2*x^32 + 5*x^33 + 6*x^34 + 7*x^35 + 4*x^36 + 6*x^37 + 4*x^39 + 4*x^40 + 4*x^41 + 6*x^43 + 3*x^47 + 2*x^48 + 2*x^49 + 2*x^51 + 3*x^55 + x^63 + 2*x^64 + 2*x^65 + 2*x^67 + 2*x^71 + 2*x^79 + x^95 + x^127)*A(x)^7 +...

%e which is a series generated by the continued fraction expression.

%o (PARI) {a(n)=local(A=1+x, CF,M=#binary(n)); for(i=1, n, CF=1+x; for(k=0,M, CF=1/(1-x^(2^(M-k))*A*CF+x*O(x^n))); A=CF); polcoeff(A, n)}

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

%Y Cf. A192739.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 08 2011