A187001 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^3 *x^k* A(x)^(3k)].

1, 1, 2, 12, 62, 300, 1635, 9413, 54505, 321655, 1938621, 11834305, 72975115, 454385175, 2852742151, 18034439209, 114709370133, 733605250447, 4714351916849, 30426720332601, 197141722168571, 1281835943761551

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

G.f. satisfies:

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

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

Example

 G.f.: A(x) = 1 + x + 2*x^2 + 12*x^3 + 62*x^4 + 300*x^5 + 1635*x^6 +...
where g.f. A(x) satisfies:
* A(x) = 1 + x*(1 + x*A(x)^3) + x^2*(1 + 8*x*A(x)^3 + x^2*A(x)^6) + x^3*(1 + 27*x*A(x)^3 + 27*x^2*A(x)^6 + x^3*A(x)^9) + x^4*(1 + 64*x*A(x)^3 + 216*x^2*A(x)^6 + 64*x^3*A(x)^9 + x^4*A(x)^12) +...;
* A(x) = 1/(1-x-x^2*A(x)^3) + 6*x^3*A(x)^3/(1-x-x^2*A(x)^3)^4 + 90*x^6*A(x)^6/(1-x-x^2*A(x)^3)^7 + 1680*x^9*A(x)^9/(1-x-x^2*A(x)^3)^10 + 34650*x^12*A(x)^12/(1-x-x^2*A(x)^3)^13 +...

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))^(3*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))^(3*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^(3*m)/(1-x-x^2*A^3+x*O(x^n))^(3*m+1))); polcoeff(A, n)}

Crossrefs

Cf. A186097, A187000.

Sequence in context: A037562 A321277 A377464 * A121177 A125831 A289787

Adjacent sequences: A186998 A186999 A187000 * A187002 A187003 A187004

Keyword

nonn

Author

Paul D. Hanna, Mar 01 2011

Status

approved