A187000 G.f. satisfies: A(x) = Sum_{n>=0} x^n*[Sum_{k=0..n} C(n,k)^3 *x^k* A(x)^(2k)].
1, 1, 2, 11, 50, 216, 1062, 5457, 28088, 147686, 792716, 4303036, 23587816, 130552984, 728316462, 4090107939, 23106799662, 131234374020, 748839982332, 4290960328482, 24681334371228, 142454368143300, 824786385860748
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 50*x^4 + 216*x^5 + 1062*x^6 +... where g.f. A(x) satisfies: * A(x) = 1 + x*(1 + x*A(x)^2) + x^2*(1 + 8*x*A(x)^2 + x^2*A(x)^4) + x^3*(1 + 27*x*A(x)^2 + 27*x^2*A(x)^4 + x^3*A(x)^6) + x^4*(1 + 64*x*A(x)^2 + 216*x^2*A(x)^4 + 64*x^3*A(x)^6 + x^4*A(x)^8) +...; * A(x) = 1/(1-x-x^2*A(x)^2) + 6*x^3*A(x)^2/(1-x-x^2*A(x)^2)^4 + 90*x^6*A(x)^4/(1-x-x^2*A(x)^2)^7 + 1680*x^9*A(x)^6/(1-x-x^2*A(x)^2)^10 + 34650*x^12*A(x)^8/(1-x-x^2*A(x)^2)^13 +...
Programs
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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))^(2*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^(2*m)/(1-x-x^2*A^2+x*O(x^n))^(3*m+1))); polcoeff(A, n)}
Formula
G.f. satisfies:
(1) A(x) = Sum_{n>=0} x^(2n)*A(x)^(2n)*[Sum_{k>=0} C(n+k,k)^3*x^k].
(2) A(x) = Sum_{n>=0} (3n)!/n!^3 * x^(3n)*A(x)^(2n)/(1-x-x^2*A(x)^2)^(3n+1).