A200475 G.f. satisfies: A(x) = exp( Sum_{n>=1} (Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^(2*k)) * x^n*A(x)^n/n ), where A027907 is the triangle of trinomial coefficients.
1, 1, 3, 13, 65, 350, 1981, 11627, 70132, 432090, 2707595, 17202779, 110563543, 717547090, 4695774335, 30952628861, 205318395288, 1369539030021, 9180527051187, 61813112864984, 417850301293691, 2834802846097200, 19294989810689802, 131723105933867817, 901709774424393614
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 65*x^4 + 350*x^5 + 1981*x^6 +... Let A = g.f. A(x), then the logarithm of the g.f. equals the series: log(A(x)) = (1 + x*A^2 + x^2*A^4)*x*A + (1 + 2^2*x*A^2 + 3^2*x^2*A^4 + 2^2*x^3*A^6 + x^4*A^8)*x^2*A^2/2 + (1 + 3^2*x*A^2 + 6^2*x^2*A^4 + 7^2*x^3*A^6 + 6^2*x^4*A^8 + 3^2*x^5*A^10 + x^6*A^12)*x^3*A^3/3 + (1 + 4^2*x*A^2 + 10^2*x^2*A^4 + 16^2*x^3*A^6 + 19^2*x^4*A^8 + 16^2*x^5*A^10 + 10^2*x^6*A^12 + 4^2*x^7*A^14 + x^8*A^16)*x^4*A^4/4 + (1 + 5^2*x*A^2 + 15^2*x^2*A^4 + 30^2*x^3*A^6 + 45^2*x^4*A^8 + 51^2*x^5*A^10 + 45^2*x^6*A^12 + 30^2*x^7*A^14 + 15^2*x^8*A^16 + 5^2*x^9*A^18 + x^10*A^20)*x^5*A^5/5 +... which involves the squares of the trinomial coefficients A027907(n,k).
Programs
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PARI
{a(n)=local(A=1+x);for(i=1,n,A=(1-x*A^2+x^3*A^6-x^5*A^10+x^6*A^12)/(1-x*A^2+x*O(x^n))^2);polcoeff(A,n)} -
PARI
/* G.f. A(x) using the squares of the trinomial coefficients */ {A027907(n, k)=polcoeff((1+x+x^2)^n, k)} {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(k=0, 2*m, A027907(m, k)^2 *x^k*(A+x*O(x^n))^(2*k))*x^m*A^m/m))); polcoeff(A, n)}
Formula
G.f. satisfies: A(x) = (1 + x^3*A(x)^6)*(1 + x^6*A(x)^12)/((1 - x*A(x)^2)*(1 - x^4*A(x)^8)).
Comments