A259269 - OEIS

%I #7 Aug 24 2017 10:45:49

%S 1,1,12,264,7858,282972,11675841,535230939,26735073957,1436236487580,

%T 82211207568861,4979654512195446,317494071032079069,

%U 21219516654529825396,1481652170309786445597,107788957126134284934186,8151161821017797142225705,639483016955485718843031996

%N G.f. A(x) satisfies: A'(x)/2 = Series_Reversion( x - 3*A(x)^2 * A'(x)/2 ).

%H Vaclav Kotesovec, <a href="/A259269/b259269.txt">Table of n, a(n) for n = 1..180</a>

%F G.f. A(x) satisfies:

%F (1) A'(x) = 2*x + 2*Sum_{n>=1} d^(n-1)/dx^(n-1) (3/2)^n * A(x)^(2*n) * A'(x)^n / n!.

%F (2) A'(x) = 2*x * exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (3/2)^n * A(x)^(2*n) * A'(x)^n / (n!*x) ).

%F a(n) = A259268(n) / (2*n-1).

%F a(n) == 1 (mod 3) iff (2*n-1) is a power of 3, and a(n) == 0 (mod 3) elsewhere (conjecture).

%e G.f.: A(x) = x^2 + x^6 + 12*x^10 + 264*x^14 + 7858*x^18 + 282972*x^22 +...

%e Let G(x) be the g.f. of A259268 such that

%e G(x) = A'(x)/2 = x + 3*x^5 + 60*x^9 + 1848*x^13 + 70722*x^17 + 3112692*x^21 +...+ A259268(n)*x^(4*n-3) +...

%e then G( x - 3*A(x)^2*G(x) ) = x.

%e Also,

%e A'(x)/2 = x + (3/2)*A(x)^2*A'(x) + [d/dx (3/2)^2*A(x)^4*A'(x)^2]/2! + [d^2/dx^2 (3/2)^3*A(x)^6*A'(x)^3]/3! + [d^3/dx^3 (3/2)^4*A(x)^8*A'(x)^4]/4! + [d^4/dx^4 (3/2)^5*A(x)^10*A'(x)^5]/5! +...

%o (PARI) {a(n)=local(A=x^2,G=x); for(i=0, n, A=intformal(2*G);  G = serreverse(x - 3*A^2*G +O(x^(4*n)))); polcoeff(A, 4*n-2)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=x^2, G=x); for(i=1, n, A=intformal(2*G);  G = serreverse(x - 3*A^2*G +O(x^(4*n)))); polcoeff(A, 4*n-2)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x^2); for(i=1, n, A = 2*intformal(x + sum(m=1, n+1, Dx(m-1, (3/2)^m*A^(2*m)*(A')^m/m!)) +O(x^(4*n)))); polcoeff(A, 4*n-2)}

%o for(n=1, 25, print1(a(n), ", "))

%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}

%o {a(n)=local(A=x^2); for(i=1, n, A = 2*intformal(x*exp(sum(m=1, n, Dx(m-1, (3/2)^m*A^(2*m)*(A')^m/(m!*x))) +O(x^(4*n))))); polcoeff(A, 4*n-2)}

%o for(n=1, 25, print1(a(n), ", "))

%Y Cf. A259268, A259270, A259271.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jun 30 2015