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A374571 Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2.

%I A374571 #33 Jul 15 2024 10:18:24
%S A374571 1,-1,-1,2,-1,1,2,-6,-1,5,1,0,2,-8,-6,22,-1,-11,5,-30,1,33,0,0,2,-16,
%T A374571 -8,52,-6,-40,22,-114,-1,125,-11,90,5,-149,-30,154,1,-123,33,-360,0,
%U A374571 552,0,144,2,-440,-16,256,-8,-360,52,-552,-6,1176,-40,576,22,-1360,-114,470,-1,-235,125,-1710,-11,3387,90,486,5,-3353,-149,1864,-30,-2152,154,-2250,1
%N A374571 Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2.
%C A374571 Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's.
%C A374571 Conjectures. For n > 0, we have the following occurrences:
%C A374571 a(n) = 0 iff n = 11 * 2^k or n = 23 * 2^k,
%C A374571 a(n) = 1 iff n = 5 * 2^k,
%C A374571 a(n) = 2 iff n = 3 * 2^k,
%C A374571 a(n) = 5 iff n = 9 * 2^k,
%C A374571 a(n) = 22 iff n = 15 * 2^k,
%C A374571 a(n) = 33 iff n = 21 * 2^k,
%C A374571 a(n) = 42 iff n = 131 * 2^k,
%C A374571 a(n) = 52 iff n = 27 * 2^k,
%C A374571 a(n) = 90 iff n = 35 * 2^k,
%C A374571 a(n) = 125 iff n = 33 * 2^k,
%C A374571 a(n) = 144 iff n = 47 * 2^k,
%C A374571 a(n) = 154 iff n = 39 * 2^k,
%C A374571 a(n) = 256 iff n = 51 * 2^k,
%C A374571 a(n) = 470 iff n = 63 * 2^k,
%C A374571 a(n) = -1 iff n = 2^k,
%C A374571 a(n) = -6 iff n = 7 * 2^k,
%C A374571 a(n) = -8 iff n = 13 * 2^k,
%C A374571 a(n) = -11 iff n = 17 * 2^k,
%C A374571 a(n) = -16 iff n = 25 * 2^k,
%C A374571 a(n) = -30 iff n = 19 * 2^k,
%C A374571 a(n) = -40 iff n = 29 * 2^k,
%C A374571 a(n) = -114 iff n = 31 * 2^k,
%C A374571 a(n) = -123 iff n = 41 * 2^k,
%C A374571 a(n) = -149 iff n = 37 * 2^k,
%C A374571 a(n) = -235 iff n = 65 * 2^k,
%C A374571 a(n) = -360 iff n = 43 * 2^k or n = 53 * 2^k,
%C A374571 etc., each of which hold for k >= 0.
%H A374571 Paul D. Hanna, <a href="/A374571/b374571.txt">Table of n, a(n) for n = 0..5000</a>
%F A374571 G.f. A(x) = Sum_{n>=0} a(n)*x^n, where B(x) is the g.f. of A374570 and C(x) = x + C(x)^2 is the g.f. of A000108, satisfies the following formulas.
%F A374571 (1) A(x) = A(x^2) - x*A(x^2)^2.
%F A374571 (2) A(x^2) = (1 - sqrt(1 - 4*x*A(x))) / (2*x).
%F A374571 (3) A(x^2) = (1/x) * C(x*A(x)).
%F A374571 (4) x^2 = B( x * C(x*A(x)) ).
%F A374571 (5) A(B(x)) = x / B(x).
%F A374571 (6) A(B(x)^2) = C(x) / B(x).
%F A374571 (7) B(x)^2 = B( B(x)*C(x) ).
%e A374571 G.f.: A(x) = 1 - x - x^2 + 2*x^3 - x^4 + x^5 + 2*x^6 - 6*x^7 - x^8 + 5*x^9 + x^10 + 2*x^12 - 8*x^13 - 6*x^14 + 22*x^15 - x^16 - 11*x^17 + 5*x^18 - 30*x^19 + x^20 + ...
%e A374571 where A(x^2) = (1 - sqrt(1 - 4*x*A(x)))/(2*x).
%e A374571 RELATED SERIES.
%e A374571 Let B(x) = Series_Reversion(x*A(x)), then
%e A374571 B(x) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 90*x^6 + 320*x^7 + 1152*x^8 + 4257*x^9 + 15934*x^10 + 60486*x^11 + 231894*x^12 + ... + A374570(n)*x^n + ...
%e A374571 where B(x)^2 = B( B(x)*C(x) ), and C(x) begins:
%e A374571 C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + ... + A000108(n)*x^n + ,,,
%e A374571 where C(x) = (1 - sqrt(1 - 4*x))/2 is the Catalan function.
%e A374571 SPECIFIC VALUES.
%e A374571 A(t) = 4/5 at t = 0.1786763406278486221896028296025274247659944115...
%e A374571 A(t) = 3/4 at t = 0.2209727374872302749773868295900473238254186343...
%e A374571 A(t) = 2/3 at t = 0.2927920532546611624693565662579476873870699464...
%e A374571 A(t) = 3/5 at t = 0.3532836501852252091389612952989266014287213872...
%e A374571 A(t) = 1/2 at t = 0.4540878993396162878365437853450173746622109652...
%e A374571 A(t) = 2/5 at t = 0.5753264646036491718800481741299163550606457682...
%e A374571 A(t) = 1/3 at t = 0.6711059159867924708010090309770441047524321152...
%e A374571 A(t) = 1/4 at t = 0.8063263233032142016966341297674341884930955548...
%e A374571 A(t) = 1/5 at t = 0.8884702348196434968520432792716046325517863531...
%e A374571 A(1/2) = 0.4596569887547343191321148479065626411948116168891503813...
%e A374571 where A(1/4) = (1 - sqrt(1 - 2*A(1/2))).
%e A374571 A(1/3) = 0.6215166290026409046430206750366100166629591510407086872...
%e A374571 where A(1/9) = (3/2) * (1 - sqrt(1 - (4/3)*A(1/3))).
%e A374571 A(1/4) = 0.7159471484203487850228006105062270686816491955635126263...
%e A374571 where A(1/16) = 2 * (1 - sqrt(1 - A(1/4))).
%e A374571 A(1/5) = 0.7747713037551020088783260174094983351988173792698848600...
%e A374571 where A(1/25) = (5/2) * (1 - sqrt(1 - (4/5)*A(1/5))).
%e A374571 A(1/6) = 0.8141931617547219509824463958597943246122338043286847588...
%e A374571 where A(1/36) = 3 * (1 - sqrt(1 - (2/3)*A(1/6))).
%e A374571 A(1/8) = 0.8630723739180924020163457579861333293488991044015651008...
%e A374571 where A(1/64) = 4 * (1 - sqrt(1 - (1/2)*A(1/8))).
%e A374571 A(1/10) = 0.891911395101161792043000371010714789952867553398091597...
%e A374571 where A(1/100) = 5 * (1 - sqrt(1 - (2/5)*A(1/10))).
%o A374571 (PARI) {a(n) = my(A = 1+x); for(i=0,#binary(n), A = subst(A,x,x^2) - x*subst(A^2,x,x^2) + x*O(x^n) ); polcoeff(A,n)}
%o A374571 for(n=0, 80, print1(a(n), ", "))
%Y A374571 Cf. A073711, A374570, A000108, A003714.
%K A374571 sign
%O A374571 0,4
%A A374571 _Paul D. Hanna_, Jul 11 2024