A229116 G.f.: A(x) = exp( Sum_{n>=1} A((-1)^n*x)^n * x^n/n ).

1, 1, 0, 1, 0, 2, 2, 9, 8, 38, 28, 154, 126, 676, 602, 3129, 2816, 14718, 13384, 70334, 65204, 342108, 321788, 1686698, 1602214, 8402492, 8051652, 42239764, 40797750, 214045640, 208136494, 1092138905, 1068176200, 5606018286, 5511336912, 28929594902, 28571895096, 150000016044

Offset: 0

Author

Paul D. Hanna, Sep 14 2013

Keywords

• nonn

Comments

Compare to a g.f. involving the Catalan function C(x) = 1 + x*C(x)^2 (A000108):

C(x) = exp( Sum_{n>=1} C(x)^n * x^n/n ).

Examples

			G.f.: A(x) = 1 + x + x^3 + 2*x^5 + 2*x^6 + 9*x^7 + 8*x^8 + 38*x^9 +...
where
log(A(x)) = A(-x)*x + A(x)^2*x^2/2 + A(-x)^3*x^3/3 + A(x)^4*x^4/4 + A(-x)^5*x^5/5 + A(x)^6*x^6/6 + A(-x)^7*x^7/7 +...
Also,
A(x)*(1 + x*A(x)) = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 4*x^5 + 6*x^6 + 18*x^7 + 30*x^8 + 76*x^9 + 124*x^10 + 308*x^11 + 514*x^12 +...
where 1/(A(x)*(1 + x*A(x))) = A(-x)*(1 - x*A(-x)).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..500
• Vaclav Kotesovec, Recurrence (of order 14)

Crossrefs

Cf. A157674.

Programs

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(k=1, n, subst(A, x, (-1)^k*x+x*O(x^n))^k*x^k/k))); polcoeff(A, n)}
for(n=0,50,print1(a(n),", "))

PARI

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=sqrt( (1 + x*subst(A,x,-x))/((1 - x*subst(A,x,-x))*(1 - x^2*A^2)) +x*O(x^n))); polcoeff(A, n)}
for(n=0,50,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = sqrt( (1 - x^2*A(-x)^2)/(1 - x^2*A(x)^2) ) / (1 - x*A(-x)).

(2) A(x) = 1/( (1 + x*A(x)) * A(-x) * (1 - x*A(-x)) ).

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

(4) A(x) = 1/(2*A(-x)*(1 - x*A(-x))) + A(-x)*(1 + x*A(-x))/2.

a(n) ~ c * d^n/(sqrt(Pi)*n^(3/2)), where d = sqrt((37 + (182701 - 19488*sqrt(87))^(1/3) + (182701 + 19488*sqrt(87))^(1/3))/21) = 2.37234975879070748... is the root of the equation -256 + 32*d^2 - 37*d^4 + 7*d^6 = 0. If n is even then c = sqrt((522 - 19*174^(2/3)/(92133 - 9877*sqrt(87))^(1/3) - (174*(92133 - 9877*sqrt(87)))^(1/3))/1479) = 0.3620905463490063953... is the root of the equation 182*c^2 - 522*c^4 + 493*c^6 = 16. If n is odd then c = sqrt(((58*(29 - 3*sqrt(87)))^(1/3) + (58*(29 + 3*sqrt(87)))^(1/3))/29) = 0.8049267655440167596... is the root of the equation 29*c^6 - 6*c^2 = 4. - Vaclav Kotesovec, Sep 15 2013