OFFSET
0,3
COMMENTS
Not the same as A179486; the first term that differs is 401708635.
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. [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). [Paul D. Hanna, Jun 24 2011]
a(n+1) = Sum_{k=0..floor(n/3)} binomial(n,k)*binomial(3*n-3*k+3,n-3*k)/(n-k+1). - Emanuele Munarini, Jun 20 2024
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.
MATHEMATICA
Table[Sum[Binomial[n, k] Binomial[3n-3k+3, n-3k]/(n-k+1), {k, 0, n/3}], {n, 0, 100}] (* for a(n+1) *) (* Emanuele Munarini, Jun 20 2024 *)
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, Jun 24 2011 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Kruchinin, Jun 24 2011
STATUS
approved