A186097 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^3 *x^k* A(x)^k].

1, 1, 2, 10, 39, 147, 639, 2857, 12725, 58081, 270250, 1268444, 6009439, 28736727, 138401100, 670641714, 3268021317, 16004012529, 78716657052, 388701645264, 1926266491659, 9576792342099, 47753368809171, 238759903786041

Offset

0,3

Links

Table of n, a(n) for n=0..23.

Formula

G.f. satisfies:

(1) A(x) = Sum_{n>=0} x^(2n)*A(x)^n*[Sum_{k>=0} C(n+k,k)^3*x^k].

(2) A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(3n)*A(x)^n/(1-x-x^2*A(x))^(3n+1).

Example

G.f.: A(x) = 1 + x + 2*x^2 + 10*x^3 + 39*x^4 + 147*x^5 + 639*x^6 +...
where g.f. A(x) satisfies:
* A(x) = 1 + x*(1 + x*A(x)) + x^2*(1 + 8*x*A(x) + x^2*A(x)^2) + x^3*(1 + 27*x*A(x) + 27*x^2*A(x)^2 + x^3*A(x)^3) + x^4*(1 + 64*x*A(x) + 216*x^2*A(x)^2 + 64*x^3*A(x)^3 + x^4*A(x)^4) +...;

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*sum(k=0, m, binomial(m, k)^3*x^k*(A+x*O(x^n))^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\2, x^(2*m)*(A+x*O(x^n))^m*sum(k=0, n, binomial(m+k, k)^3*x^k))); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n\3, (3*m)!/m!^3*x^(3*m)*A^m/(1-x-x^2*A+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

Crossrefs

Cf. A181665, A183876, A181543.

Sequence in context: A122117 A322211 A120949 * A295521 A165814 A151311

Adjacent sequences: A186094 A186095 A186096 * A186098 A186099 A186100

Keyword

nonn

Author

Paul D. Hanna, Feb 12 2011

Status

approved