A274717 - OEIS

%I #10 Jul 07 2016 18:12:33

%S 1,1,1,1,1,1,7,1,-21,1,71,1,5,7,1095,1,-15885,-21,18443,1,-55841,71,

%T 324945,1,-2649857,5,6109987,7,-18206579,1095,92290439,1,-3700779069,

%U -15885,5957604211,-21,-29227819205,18443,88113645133,1,-917331711003,-55841,2134630503193,71,-9943308217037,324945,29285764343377,1,-616425445539209,-2649857,1440419971225759,5,-7198783835108021,6109987,19846350459729237

%N G.f. satisfies: A(x) = A(x^2) + sqrt( A(x^2) ), where A(x) = Sum_{n>=1} a(n) * x^n / 2^A274716(n).

%H Paul D. Hanna, <a href="/A274717/b274717.txt">Table of n, a(n) for n = 1..1030</a>

%F a(2*n) = a(n) for n>=1.

%e G.f.: A(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 - 15885/32768*x^17 - 21/128*x^18 + 18443/65536*x^19 + 1/8*x^20 - 55841/262144*x^21 + 71/256*x^22 + 324945/524288*x^23 + 1/2*x^24 - 2649857/4194304*x^25 + 5/1024*x^26 + 6109987/8388608*x^27 + 7/16*x^28 - 18206579/33554432*x^29 + 1095/2048*x^30 + 92290439/67108864*x^31 + x^32 +...+ a(n)*x^n/2^A274716(n) +...

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

%e RELATED SERIES.

%e The following series forms an odd function:

%e A(x) - A(x^2) = x + 1/2*x^3 + 1/8*x^5 + 7/16*x^7 - 21/128*x^9 + 71/256*x^11 + 5/1024*x^13 + 1095/2048*x^15 - 15885/32768*x^17 + 18443/65536*x^19 - 55841/262144*x^21 + 324945/524288*x^23 - 2649857/4194304*x^25 + 6109987/8388608*x^27 - 18206579/33554432*x^29 +...

%e where ( A(x) - A(x^2) )^2 = A(x^2):

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

%o (PARI) {A274716(n) = if(n<3,0, if(n%2==0, A274716(n/2), A274716(2*(n\4)+1) + n\2 ) )}

%o {a(n) = my(A=x+x^2); for(i=0,#binary(n),

%o A = subst(A,x,x^2) + sqrt( subst(A,x,x^2 +x^2*O(x^n)) ) );

%o 2^A274716(n)*polcoeff(A,n)}

%o for(n=1,65,print1(a(n),", "))

%Y Cf. A274716.

%K sign

%O 1,7

%A _Paul D. Hanna_, Jul 07 2016