A192132 G.f. satisfies: A(x) = 1 + x*A(x)^3 + x*(A(x) - 1)^3.

1, 1, 3, 12, 56, 285, 1533, 8571, 49311, 290019, 1735845, 10538550, 64741482, 401708635, 2513837901, 15847466748, 100547969532, 641570954283, 4114313131809, 26503231512588, 171416266008912, 1112726163448431, 7247066415484731

Offset

0,3

Comments

Not the same as A179486; the first term that differs is 401708635.

Links

Table of n, a(n) for n=0..22.

Formula

a(n)=sum(i=0..n-1, 2^i*binomial(n,i)*sum(j=0..n, binomial(j,-n+2*j-i-1)*binomial(n,j)))/n, n>0, a(0)=1.

G.f. satisfies: 1 + x*A(x)^3*G( x^3*A(x)^6 ) where G(x) = 1 + x*G(x)^3 = g.f. of A001764. [From Paul D. Hanna, Jun 24 2011]

G.f. satisfies: A(x) = 1 + Sum_{n>=0} binomial(3*n+1,n)/(3*n+1) * x^(3*n+1)*A(x)^(6*n+3). [From Paul D. Hanna, Jun 24 2011]

Example

G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 285*x^5 + 1533*x^6 +...
where the g.f. satisfies:
A(x) = 1 + x*A(x)^3 + x^4*A(x)^9 + 3*x^7*A(x)^15 + 12*x^10*A(x)^21 + 55*x^13*A(x)^27 +...+ A001764(n)*x^(3*n+1)*A(x)^(6*n+3) +...
The related expansions:
A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 276*x^4 + 1470*x^5 + 8160*x^6 +...
(A(x)-1)^3 = x^3 + 9*x^4 + 63*x^5 + 411*x^6 + 2619*x^7 + 16569*x^8 +...
illustrate the property: A(x) = 1 + x*A(x)^3 + x*(A(x) - 1)^3.

Prog

(Maxima)
a(n):=if n<1 then 1 else sum(2^i*binomial(n, i)*sum(binomial(j, -n+2*j-i-1)*binomial(n, j), j, 0, n), i, 0, n-1)/n;
(PARI) {a(n)=if(n==0, 1, sum(i=0, n-1, 2^i*binomial(n, i)*sum(j=0, n, binomial(j, -n+2*j-i-1)*binomial(n, j))/n))}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x*A^3+x*(A-1+x*O(x^n))^3); polcoeff(A, n)}
(PARI) {a(n)=local(A=1+x, G=sum(m=0, n, binomial(3*m+1, m)/(3*m+1)*x^m)+x*O(x^n));
for(i=1, n, A=1+x*A^3*subst(G, x, x^3*A^6)); polcoeff(A, n)} /* Paul D. Hanna */

Crossrefs

Cf. A001764.

Sequence in context: A226316 A195261 A276902 * A179486 A369482 A366097

Adjacent sequences: A192129 A192130 A192131 * A192133 A192134 A192135

Keyword

nonn

Author

Vladimir Kruchinin, Jun 24 2011

Status

approved