A259269 G.f. A(x) satisfies: A'(x)/2 = Series_Reversion( x - 3*A(x)^2 * A'(x)/2 ).

1, 1, 12, 264, 7858, 282972, 11675841, 535230939, 26735073957, 1436236487580, 82211207568861, 4979654512195446, 317494071032079069, 21219516654529825396, 1481652170309786445597, 107788957126134284934186, 8151161821017797142225705, 639483016955485718843031996

Offset

1,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..180

Formula

G.f. A(x) satisfies:

(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!.

(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) ).

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

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

Example

G.f.: A(x) = x^2 + x^6 + 12*x^10 + 264*x^14 + 7858*x^18 + 282972*x^22 +...
Let G(x) be the g.f. of A259268 such that
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) +...
then G( x - 3*A(x)^2*G(x) ) = x.
Also,
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! +...

Prog

(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)}
for(n=1, 25, print1(a(n), ", "))
(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)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{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)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A259268, A259270, A259271.

Sequence in context: A138451 A295844 A113672 * A035265 A167869 A068271

Adjacent sequences: A259266 A259267 A259268 * A259270 A259271 A259272

Keyword

nonn

Author

Paul D. Hanna, Jun 30 2015

Status

approved