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%I A350481 #7 Sep 01 2022 12:00:02
%S A350481 1,-1,0,-1,4,-6,4,-1,0,-1,10,-45,124,-262,564,-1360,3076,-5913,10238,
%T A350481 -17861,32040,-54785,85310,-124275,178760,-256919,354870,-455046,
%U A350481 543444,-623196,702354,-775965,854874,-1083885,2001516,-5152914,14709734
%N A350481 G.f. A(x) satisfies: A(x) = A(x^3 - x^6)/x^2.
%F A350481 G.f. A(x) = Sum_{n>=1} a(n)*x^(3*n-2) satisfies:
%F A350481 (1) A(x) = A(x^3 - x^6)/x^2.
%F A350481 (2) R(x^2*A(x)) = x^3 - x^6, where R(A(x)) = x.
%F A350481 (3) A(x) = Product_{n>=0} F(n), where F(0) = x, F(1) = 1-x^3, and F(n+1) = 1 - (1 - F(n))^3 * F(n)^3 for n > 0.
%e A350481 G.f.: A(x) = x - x^4 - x^10 + 4*x^13 - 6*x^16 + 4*x^19 - x^22 - x^28 + 10*x^31 - 45*x^34 + 124*x^37 - 262*x^40 + ...
%e A350481 The series reversion is here denoted R(x) so that R(A(x)) = x where
%e A350481 R(x) = x + x^4 + 4*x^7 + 23*x^10 + 150*x^13 + 1060*x^16 + 7910*x^19 + 61319*x^22 + 488982*x^25 + ... + A350480(n)*x^(3*n-2) + ...
%e A350481 and which by definition also satisfies R(x^2*A(x)) = x^3 - x^6.
%e A350481 GENERATING METHOD.
%e A350481 One may generate the g.f. A(x) using the following method.
%e A350481 Define F(n), a polynomial in x of order 3*6^(n-1), by the following recurrence:
%e A350481 F(0) = x,
%e A350481 F(1) = (1 - x^3),
%e A350481 F(2) = (1 - x^9 * (1-x^3)^3),
%e A350481 F(3) = (1 - x^27 * (1-x^3)^9 * F(2)^3),
%e A350481 F(4) = (1 - x^81 * (1-x^3)^27 * F(2)^9 * F(3)^3),
%e A350481 F(5) = (1 - x^243 * (1-x^3)^81 * F(2)^27 * F(3)^9 * F(4)^3),
%e A350481 ...
%e A350481 F(n+1) = 1 - (1 - F(n))^3 * F(n)^3
%e A350481 ...
%e A350481 Then the g.f. A(x) equals the infinite product:
%e A350481 A(x) = x * F(1) * F(2) * F(3) * ... * F(n) * ...
%e A350481 that is,
%e A350481 A(x) = x * (1-x^3) * (1 - x^9*(1-x^3)^3) * (1 - x^27*(1-x^3)^9*(1 - x^9*(1-x^3)^3)^3) * (1 - x^81*(1-x^3)^27*(1 - x^9*(1-x^3)^3)^9*(1 - x^27*(1-x^3)^9*(1 - x^9*(1-x^3)^3)^3)^3) * ...
%e A350481 SPECIFIC VALUES.
%e A350481 The infinite product formula allows us to evaluate the function A(x) at certain x rather quickly.
%e A350481 A(1/2) = (1/2) * (7/2^3) * (261801/2^18) * (324518552934330535025670717244849/2^108) * ... = 0.43692755601666346006790651129...
%e A350481 A(2/3) = (2/3) * (19/3^3) * (383908681/3^18) * ... = 0.46488294171447073294042484530...
%e A350481 A(1/3) = (1/3) * (26/3^3) * (387402913/3^18) * ... = 0.32097309216130808472175568906...
%e A350481 The first relative maximum value of A(x) is given by
%e A350481 A(0.62382241029098620649978...) = 0.46944663707521689313693...
%o A350481 (PARI) {a(n) = my(A, R=[1,0]); for(i=1,n, R=concat(R,0);
%o A350481 R[#R] = -polcoeff( x^3*(1 - x^3) - subst(x*Ser(R),x, x^2 * serreverse(x*Ser(R))), #R+2) );
%o A350481 A=Vec(serreverse(x*Ser(R)));H=A; A[n]}
%o A350481 for(n=1,40,print1(a(3*n-2),", "))
%o A350481 (PARI) /* Using Infinite Product Formula for Series Reversion */
%o A350481 N = 300; \\ set limit on order of polynomials to be 3 times desired number of terms
%o A350481 {F(n) = my(G=x); if(n==0,G=x, if(n==1, G = (1-x^3), G = 1 - (1 - F(n-1))^3 * F(n-1)^3 +x^2*O(x^N)  ));G}
%o A350481 {a(n) = my(A = prod(k=0,#binary(n), F(k) +x*O(x^n))); polcoeff(A,n)}
%o A350481 for(n=1,40,print1(a(3*n-2),", "))
%Y A350481 Cf. A350480 (inverse), A273218, A350431, A350433, A350475, A350477, A350483.
%K A350481 sign
%O A350481 1,5
%A A350481 _Paul D. Hanna_, Jan 01 2022

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