A182267 G.f. satisfies: A(x) = (1+x*A(x))*(1+x^2*A(x)^2)*(1+x^3*A(x)).

1, 1, 2, 6, 16, 46, 140, 435, 1382, 4474, 14687, 48787, 163703, 554009, 1888794, 6481220, 22366415, 77575617, 270277602, 945480612, 3319582632, 11693824752, 41318554495, 146399071577, 520042511448, 1851657641932, 6607352892709, 23624965371264

Offset: 0

Author

Paul D. Hanna, Apr 22 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 46*x^5 + 140*x^6 + 435*x^7 +...
Related expansions:
A(x)^2 = 1 + 2*x + 5*x^2 + 16*x^3 + 48*x^4 + 148*x^5 + 472*x^6 +...
A(x)^3 = 1 + 3*x + 9*x^2 + 31*x^3 + 102*x^4 + 336*x^5 + 1124*x^6 +...
A(x)^4 = 1 + 4*x + 14*x^2 + 52*x^3 + 185*x^4 + 648*x^5 + 2272*x^6 +...
where A(x) = 1 + x*A(x) + x^2*A(x)^2 + x^3*(A(x) + A(x)^3) + x^4*A(x)^2 + x^5*A(x)^3 + x^6*A(x)^4.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A182053, A211854, A211855.

Programs

PARI

{a(n)=local(A=1+x); for(i=1, n, A=(1+x*A)*(1+x^2*A^2)*(1+x^3*A)+x*O(x^n)); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))

Formula

a(n) ~ sqrt(s*(1 + 2*r*s + 4*r^3*s + 5*r^4*s^2 + 6*r^5*s^3 + 3*r^2*(1 + s^2)) / (Pi*(1 + r^2 + 3*r*s + 3*r^3*s + 6*r^4*s^2))) / (2 * n^(3/2) * r^(n + 1/2)), where r = 0.2649675733882333627400730579639429790476557486165... and s = 2.383929237709193665917448862090331200952809331679... are roots of the system of equations (1 + r*s)*(1 + r^3*s)*(1 + r^2*s^2) = s, r*(1 + r^2 + 2*r*s + 2*r^3*s + 3*r^2*s^2 + 3*r^4*s^2 + 4*r^5*s^3) = 1. - Vaclav Kotesovec, Nov 18 2017