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
Keywords
Examples
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.
Crossrefs
Cf. A001764.
Programs
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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 *) -
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;
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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 */
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
Comments