A259270 G.f. satisfies: A(x) = Series_Reversion( x - 2*A(x)*B(x) ), where B'(x) = 2*A(x).

1, 2, 18, 244, 4090, 78636, 1670452, 38369256, 939975210, 24332874300, 661203162972, 18767610621144, 554355352005156, 16990196093594808, 539037630578944680, 17669190941910393296, 597443704506079947274, 20809975126591414324188, 745817743891030392909644, 27474840405379231845795960

Offset

1,2

Comments

Conjecture: n | a(n) for n>=1 and therefore all the terms of this sequence are integers.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..200

Formula

Let B(x) = Integral 2*A(x) dx, then g.f. A(x) satisfies:

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

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

a(n)/n = A259271(n).

Example

G.f.: A(x) = x + 2*x^3 + 18*x^5 + 244*x^7 + 4090*x^9 + 78636*x^11 +...+ a(n)*x^(2*n-1) +...
Let B(x) = Integral 2*A(x) dx, an integer series that begins:
B(x) = x^2 + x^4 + 6*x^6 + 61*x^8 + 818*x^10 + 13106*x^12 + 238636*x^14 + 4796157*x^16 + 104441690*x^18 +...+ A259271(n)*x^(2*n) +...
then A(x - 2*A(x)*B(x)) = x.
Also,
A(x) = x + 2*A(x)*B(x) + [d/dx 4*A(x)^2*B(x)^2]/2! + [d^2/dx^2 8*A(x)^3*B(x)^3]/3! + [d^3/dx^3 16*A(x)^4*B(x)^4]/4! + [d^4/dx^4 32*A(x)^5*B(x)^5]/5! +...
Logarithmic series:
log(A(x)/x) = 2*A(x)*B(x)/x + [d/dx 4*A(x)^2*B(x)^2/x]/2! + [d^2/dx^2 8*A(x)^3*B(x)^3/x]/3! + [d^3/dx^3 16*A(x)^4*B(x)^4/x]/4! + [d^4/dx^4 32*A(x)^5*B(x)^5/x]/5! +...
Explicitly,
log(A(x)/x) = 4*x^2/2 + 64*x^4/4 + 1264*x^6/6 + 28064*x^8/8 + 675504*x^10/10 + 17304544*x^12/12 + 466669536*x^14/14 +...+ A259272(n)*x^(2*n)/(2*n) +...

Prog

(PARI) {a(n)=local(A=x+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A);  A = serreverse(x - 2*A*B +O(x^(2*n)))); polcoeff(A, 2*n-1)}
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+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A);  A = x + sum(m=1, n+1, Dx(m-1, 2^m*A^m*B^m/m!)) +O(x^(2*n))); polcoeff(A, 2*n-1)}
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+x*O(x^n), B=x^2); for(i=1, n, B=intformal(2*A); A = x*exp(sum(m=1, n, Dx(m-1, 2^m*A^m*B^m/(m!*x))) +O(x^(2*n)))); polcoeff(A, 2*n-1)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A259271, A259272, A259268.

Sequence in context: A024486 A052635 A366001 * A168562 A117514 A138437

Adjacent sequences: A259267 A259268 A259269 * A259271 A259272 A259273

Keyword

nonn

Author

Paul D. Hanna, Jun 29 2015

Status

approved