This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373312 #26 Apr 13 2025 19:58:23
%S A373312 1,2,9,46,266,1636,10529,69974,476598,3309212,23336626,166686732,
%T A373312 1203409180,8767531432,64378620609,475951684454,3539801952222,
%U A373312 26466142669804,198814291126846,1499817211781796,11357495427008900,86302897747248024,657858710864911954,5029067212015246972
%N A373312 Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^2 ).
%C A373312 Conjecture: a(n) == 1 (mod 2) iff n = 2^k - 1 for k >= 1.
%H A373312 Paul D. Hanna, <a href="/A373312/b373312.txt">Table of n, a(n) for n = 1..520</a>
%F A373312 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A373312 (1) A(x)^2 = A( x*A(x)/(1 - A(x))^2 ).
%F A373312 (2) A(x)^4 = A( x*A(x)^3/((1 - A(x))^2*(1 - A(x)^2)^2) ).
%F A373312 (3) A(x)^8 = A( x*A(x)^7/((1 - A(x))^2*(1 - A(x)^2)^2*(1 - A(x)^4)^2) ).
%F A373312 (4) A(x)^(2^n) = A( x*A(x)^(2^n-1)/Product_{k=0..n-1} (1 - A(x)^(2^k))^2 ) for n > 0.
%F A373312 (5) A(x) = x / Product_{n>=0} (1 - A(x)^(2^n))^2.
%F A373312 (6) A(x) = x * Product_{n>=0} (1 + A(x)^(2^n))^(2*(n+1)).
%F A373312 (7) A(x) = Series_Reversion( x*B(x) ), where B(x) = Product_{n>=0} (1 - x^(2^n))^2 is the g.f. of A106407.
%F A373312 The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 2*2^n * A(r)^(2^n)/(1 - A(r)^(2^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(2^n))^2, where r = 0.1226755376200489039634571399751738538057851136283925... and A(r) = 0.2554610761187220452590974770035518728586331311247120...
%F A373312 Given r and A(r) above, A(r) also satisfies 1 = Sum_{n>=0} 2*(n+1)*2^n * A(r)^(2^n)/(1 + A(r)^(2^n)).
%e A373312 G.f.: A(x) = x + 2*x^2 + 9*x^3 + 46*x^4 + 266*x^5 + 1636*x^6 + 10529*x^7 + 69974*x^8 + 476598*x^9 + 3309212*x^10 + ...
%e A373312 where A( x*A(x)/(1 - A(x))^2 ) = A(x)^2.
%e A373312 RELATED SERIES.
%e A373312 A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 128*x^5 + 797*x^6 + 5164*x^7 + 34506*x^8 + 235984*x^9 + 1643882*x^10 + ...
%e A373312 Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
%e A373312 R(x) = x * Product_{n>=0} (1 - x^(2^n))^2 = x - 2*x^2 - x^3 + 4*x^4 - 3*x^5 + 2*x^6 + 3*x^7 - 8*x^8 + x^9 + 6*x^10 + ... + A106407(n-1) * x^n + ...
%e A373312 thus,
%e A373312 x = A(x) * (1 - A(x))^2 * (1 - A(x)^2)^2 * (1 - A(x)^4)^2 * (1 - A(x)^8)^2 * (1 - A(x)^16)^2 * ... * (1 - A(x)^(2^n))^2 * ...
%e A373312 Also, notice that the square root of A(x)/x is an integral series
%e A373312 (A(x)/x)^(1/2) = 1 + x + 4*x^2 + 19*x^3 + 106*x^4 + 636*x^5 + 4024*x^6 + 26405*x^7 + 178096*x^8 + 1227018*x^9 + 8598424*x^10 + ...
%e A373312 SPECIFIC VALUES.
%e A373312 A(t) = 1/5 at t = (1/5) * Product_{n>=0} (1 - 1/5^(2^n))^2 = 0.1175870125805304806733576532618445158357121658...
%e A373312 A(t) = 1/6 at t = (1/6) * Product_{n>=0} (1 - 1/6^(2^n))^2 = 0.1092311136132535692899568885022954464596243049...
%e A373312 A(1/9) = 0.17288740832245782814001741323630181133096513764543378...
%e A373312 A(1/10) = 0.1413215396171684711943139566840401836123301177323661...
%e A373312 A(1/11) = 0.1213541857717280074895334383318404648498876032468172...
%o A373312 (PARI) {a(n) = my(A = serreverse(x*prod(k=0,#binary(n), (1 - x^(2^k) + x*O(x^n))^2))); polcoeff(A,n)}
%o A373312 for(n=1,30,print1(a(n),", "))
%o A373312 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A373312 A[#A] = polcoeff( subst(Ser(A),x, x*Ser(A)/(1 - Ser(A))^2 ) - Ser(A)^2,#A)); A[n+1]}
%o A373312 for(n=1,30,print1(a(n),", "))
%Y A373312 Cf. A372530, A373313, A106407.
%K A373312 nonn
%O A373312 1,2
%A A373312 _Paul D. Hanna_, Jun 25 2024