A372531 - OEIS

%I #9 May 14 2024 12:16:26

%S 1,1,2,6,19,63,220,795,2942,11100,42547,165204,648423,2568522,

%T 10255044,41226054,166732446,677922831,2769487183,11362238976,

%U 46794199487,193387049685,801742251778,3333468469185,13896609698686,58073938493679,243238872937589,1020921149848044

%N Expansion of g.f. A(x) satisfying A(x)^3 = A( x*A(x)^2/(1 - A(x)) ).

%H Paul D. Hanna, <a href="/A372531/b372531.txt">Table of n, a(n) for n = 1..730</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x)^3 = A( x*A(x)^2/(1 - A(x)) ).

%F (2) A(x)^9 = A( x*A(x)^8/((1 - A(x))*(1 - A(x)^3)) ).

%F (3) A(x)^27 = A( x*A(x)^26/((1 - A(x))*(1 - A(x)^3)*(1 - A(x)^9)) ).

%F (4) A(x)^(3^n) = A( x*A(x)^(3^n-1)/Product_{k=0..n-1} (1 - A(x)^(3^k)) ) for n > 0.

%F (5) A(x) = x / Product_{n>=0} (1 - A(x)^(3^n)).

%F (6) A(x) = Series_Reversion( x * Product_{n>=0} (1 - x^(3^n)) ).

%F (7) x = A(x) * Sum_{n>=1} (-1)^A010060(n-1) * A(x)^A005836(n), where A010060 is the Thue-Morse sequence and A005836 lists numbers whose base-3 representation contains no 2.

%F The radius of convergence r and A(r) satisfy 1 = Sum_{n>=0} 3^n * A(r)^(3^n)/(1 - A(r)^(3^n)) and r = A(r) * Product_{n>=0} (1 - A(r)^(3^n)), where r = 0.225516184149820697566779292359148589008135334130979... and A(r) = 0.426459850024068213581384788221931983633117762268826...

%e G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 19*x^5 + 63*x^6 + 220*x^7 + 795*x^8 + 2942*x^9 + 11100*x^10 + 42547*x^11 + 165204*x^12 + ...

%e where A( x*A(x)^2/(1 - A(x)) ) = A(x)^3.

%e RELATED SERIES.

%e A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 111*x^7 + 405*x^8 + 1511*x^9 + 5742*x^10 + 22131*x^11 + 86310*x^12 + ...

%e Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then

%e R(x) = x * Product_{n>=0} (1 - x^(3^n)) = x - x^2 - x^4 + x^5 - x^10 + x^11 + x^13 - x^14 - x^28 + x^29 + x^31 - x^32 + x^37 - x^38 - x^40 + x^41 - x^82 + ... + (-1)^A010060(n-1)*x^(A005836(n) + 1) + ...

%e thus,

%e x = A(x) * (1 - A(x)) * (1 - A(x)^3) * (1 - A(x)^9) * (1 - A(x)^27) * (1 - A(x)^81) * ... * (1 - A(x)^(3^n)) * ...

%e SPECIFIC VALUES.

%e A(t) = 2/5 at t = (2/5) * Product_{n>=0} (1 - (2/5)^(3^n)) = 0.224581111967794306351236678951339491766788581...

%e A(t) = 1/3 at t = (1/3) * Product_{n>=0} (1 - 1/3^(3^n)) = 0.213980897639074346024397964153942364246900732...

%e A(t) = 1/4 at t = (1/4) * Product_{n>=0} (1 - 1/4^(3^n)) = 0.184569608420133580452570741795926229187208705...

%e A(1/5) = 0.2875735682779125398024437286065851185781152551441974155...

%e A(1/6) = 0.2142049226274852453913309509157678575015367219972692281...

%e A(1/7) = 0.1738274438474723142423128822604443845966959546404795042...

%e A(1/6)^3 = A(t) at t = (1/6)*A(1/6)^2/(1 - A(1/6)) = 0.0097319157371...

%e A(1/7)^3 = A(t) at t = (1/7)*A(1/7)^2/(1 - A(1/7)) = 0.0052247784954...

%o (PARI) /* From Series_Reversion( x * Product_{n>=0} (1 - x^(3^n)) ) */

%o {a(n) = my(A, M=ceil(log(n+1)/log(3))); A = serreverse( x * prod(m=0,M, 1 - x^(3^m)) + x*O(x^n) ); polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* From A(x)^3 = A( x*A(x)^2/(1 - A(x)) ) */

%o {a(n) = my(A=[0, 1],F); for(i=1, n, A = concat(A, 0); F=Ser(A);

%o A[#A] = polcoeff( subst(F, x, x*F^2/(1 - F) ) - F^3, #A+1) ); H=A; A[n+1]}

%o for(n=1, 30, print1(a(n), ", "))

%Y Cf. A372530, A010060, A005836.

%K nonn

%O 1,3

%A _Paul D. Hanna_, May 14 2024