A199248 - OEIS

%I #10 Mar 30 2012 18:37:32

%S 1,1,2,6,20,69,248,923,3523,13706,54152,216710,876607,3578405,

%T 14722432,60986158,254145337,1064712328,4481577078,18943753140,

%U 80381689202,342254333393,1461864544896,6262021627055,26894816382199,115792035533779,499648608539714,2160504474956390

%N G.f. satisfies: A(x) = exp( Sum_{n>=1} [Sum_{k=0..2*n} A027907(n,k)^2 * x^k * A(x)^k]* x^n/n ), where A027907 is the triangle of trinomial coefficients.

%C Trinomial coefficients satisfy: Sum_{k=0..2*n} A027907(n,k)*x^k = (1+x+x^2)^n.

%F G.f. satisfies: A(x) = G(x*A(x)) where A(x/G(x)) = G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2.

%F G.f.: A(x) = (1/x)*Series_Reversion( x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12) ).

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 20*x^4 + 69*x^5 + 248*x^6 + 923*x^7 +...

%e such that A(x) = G(x*A(x)) where G(x) is given by:

%e G(x) = (1 - x + x^2)*(1 - x^2 + x^4)/(1-x)^2 = (1-x^5)/(1-x) + x^3/(1-x)^2:

%e G(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 4*x^6 + 5*x^7 + 6*x^8 + 7*x^9 +...

%e ...

%e Let A = x*A(x), then the logarithm of the g.f. A(x) equals the series:

%e log(A(x)) = (1 + A + A^2)*x +

%e (1 + 2^2*A + 3^2*A^2 + 2^2*A^3 + A^4)*x^2/2 +

%e (1 + 3^2*A + 6^2*A^2 + 7^2*A^3 + 6^2*A^4 + 3^2*A^5 + A^6)*x^3/3 +

%e (1 + 4^2*A + 10^2*A^2 + 16^2*A^3 + 19^2*A^4 + 16^2*A^5 + 10^2*A^6 + 4^2*A^7 + A^8)*x^4/4 +

%e (1 + 5^2*A + 15^2*A^2 + 30^2*A^3 + 45^2*A^4 + 51^2*A^5 + 45^2*A^6 + 30^2*A^7 + 15^2*A^8 + 5^2*A^9 + A^10)*x^5/5 +...

%e which involves the squares of the trinomial coefficients A027907(n,k).

%o (PARI) {a(n)=local(A=1+x); A=1/x*serreverse(x*(1-x)*(1-x^3)*(1-x^4)/(1-x^12+x*O(x^n))); polcoeff(A, n)}

%o (PARI) /* G.f. A(x) using the squares of the trinomial coefficients */

%o {A027907(n, k)=polcoeff((1+x+x^2)^n, k)}

%o {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^k) *x^m/m)+x*O(x^n)));polcoeff(A, n)}

%Y Cf. A186236, A199257, A027907.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2011