A192132 - OEIS

%I #22 Jun 20 2024 16:33:03

%S 1,1,3,12,56,285,1533,8571,49311,290019,1735845,10538550,64741482,

%T 401708635,2513837901,15847466748,100547969532,641570954283,

%U 4114313131809,26503231512588,171416266008912,1112726163448431,7247066415484731

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

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

%F 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.

%F 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]

%F 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]

%F 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

%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 285*x^5 + 1533*x^6 +...

%e where the g.f. satisfies:

%e 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) +...

%e The related expansions:

%e A(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 276*x^4 + 1470*x^5 + 8160*x^6 +...

%e (A(x)-1)^3 = x^3 + 9*x^4 + 63*x^5 + 411*x^6 + 2619*x^7 + 16569*x^8 +...

%e illustrate the property: A(x) = 1 + x*A(x)^3 + x*(A(x) - 1)^3.

%t 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 *)

%o (Maxima)

%o 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;

%o (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))}

%o (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)}

%o (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));

%o 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 */

%Y Cf. A001764.

%K nonn

%O 0,3

%A _Vladimir Kruchinin_, Jun 24 2011