A216116 - OEIS

%I #9 Oct 29 2012 18:52:28

%S 1,1,1,1,2,4,7,11,17,28,49,87,152,262,453,794,1408,2507,4462,7943,

%T 14179,25415,45713,82398,148731,268859,486890,883411,1605582,2922259,

%U 5325377,9716564,17750332,32464980,59443403,108951953,199886003,367052947,674620772,1240963218

%N  G.f. satisfies: A(x) = (1 + x*A(x)) * (1 + x^4*A(x)).

%F  G.f.: A(x) = exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} C(n,k)^2 * x^(3*k) ).

%F G.f.: A(x) = ((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5))/(2*x^5).

%F a(n) = A023427(n+1) for n>=0.

%e  A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 4*x^5 + 7*x^6 + 11*x^7 + 17*x^8 + 28*x^9 +...

%e The logarithm of the g.f. equals the series:

%e log(A(x)) = (1 + x^3)*x +

%e (1 + 2^2*x^3 + x^6)*x^2/2 +

%e (1 + 3^2*x^3 + 3^2*x^6 + x^9)*x^3/3 +

%e (1 + 4^2*x^3 + 6^2*x^6 + 4^2*x^9 + x^12)*x^4/4 +

%e (1 + 5^2*x^3 + 10^2*x^6 + 10^2*x^9 + 5^2*x^12 + x^15)*x^5/5 +

%e (1 + 6^2*x^3 + 15^2*x^6 + 20^2*x^9 + 15^2*x^12 + 6^2*x^15 + x^18)*x^6/6 +...

%o  (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1+x*A)*(1+x^4*A)+x*O(x^n)); polcoeff(A, n)}

%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sum(j=0, m, binomial(m, j)^2*(x+x*O(x^n))^(3*j))*x^m/m)), n)}

%o (PARI) {a(n)=polcoeff(((1-x-x^4) - sqrt((1-x-x^4)^2 - 4*x^5 +x^6*O(x^n)))/(2*x^5), n)}

%o for(n=0,45,print1(a(n),", "))

%Y  Cf. A023427, A023432, A004148.

%K nonn

%O 0,5

%A _Paul D. Hanna_, Oct 29 2012