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 A229808 G.f. A(x) satisfies: A(x) = x + 2*B(x)*C(x), where B(x) = x + 3*A(x)*C(x) and C(x) = x + 5*A(x)*B(x). 3

%I #11 Nov 15 2013 19:50:35

%S 1,2,16,122,1096,10322,102856,1054322,11104336,119175842,1299844576,

%T 14360860202,160403229016,1808192266322,20545886851096,

%U 235069897341122,2705792876561056,31312112470672322,364079737402597936,4251402314343575642,49835252820257276776

%N G.f. A(x) satisfies: A(x) = x + 2*B(x)*C(x), where B(x) = x + 3*A(x)*C(x) and C(x) = x + 5*A(x)*B(x).

%F G.f. A = A(x) satisfies:

%F (1) A = x + 2*x^2*(1+3*A)*(1+5*A)/(1-15*A^2)^2.

%F (2) A = x*(1+2*B)/(1-10*B^2) where B = x*(1+3*A)/(1-15*A^2) is the g.f. of A229809.

%F (3) A = x*(1+2*C)/(1-6*C^2) where C = x*(1+5*A)/(1-15*A^2) is the g.f. of A229810.

%F The g.f.s A = A(x) (A229808), B = B(x) (A229809), C = C(x) (A229810), satisfy:

%F A*B*C = (A^2 - x*A)/2 = (B^2 - x*B)/3 = (C^2 - x*C)/5.

%e G.f.: A(x) = x + 2*x^2 + 16*x^3 + 122*x^4 + 1096*x^5 + 10322*x^6 +...

%e Related series:

%e B(x) = x + 3*x^2 + 21*x^3 + 153*x^4 + 1401*x^5 + 13083*x^6 +...

%e C(x) = x + 5*x^2 + 25*x^3 + 215*x^4 + 1825*x^5 + 17525*x^6 +...

%e where A(x) = x + 2*B(x)*C(x).

%e (A(x)^2 - x*A(x))/2 = x^3 + 10*x^4 + 93*x^5 + 920*x^6 + 9305*x^7 + 97050*x^8 + 1031737*x^9 +...

%o (PARI) {a(n)=local(A=x+x^2, B=x+2*x^2, C=x+3*x^2); for(i=1, n, A=x+2*B*C+x*O(x^n); B=x+3*A*C+x*O(x^n); C=x+5*A*B+x*O(x^n)); polcoeff(A, n)}

%o for(n=1, 30, print1(a(n), ", "))

%o (PARI) {a(n)=local(A=x);for(i=1,n,A=x+2*x^2*(1+3*A)*(1+5*A)/(1-15*A^2 +x*O(x^n))^2);polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A229809 (B(x)), A229810 (C(x)).

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 30 2013

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Last modified October 3 19:07 EDT 2023. Contains 365870 sequences. (Running on oeis4.)