A352854 - OEIS

%I #5 Apr 07 2022 12:12:25

%S 1,1,2,6,23,103,516,2819,16517,102615,670503,4580064,32553887,

%T 239884108,1827188093,14351353937,115997378072,963164672275,

%U 8203632154685,71582150287243,639150768866594,5833969369906384,54387003936658041,517419092989229133

%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n).

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

%F (1) 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n),

%F (2) 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+2)).

%e G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 23*x^4 + 103*x^5 + 516*x^6 + 2819*x^7 + 16517*x^8 + 102615*x^9 + 670503*x^10 + 4580064*x^11 + ...

%e where

%e (1) 1 = A(x) - x*A(x)^2*A(x*A(x)) + x^2*A(x)^4*A(x*A(x)^2) - x^3*A(x)^6*A(x*A(x)^3) + x^4*A(x)^8*A(x*A(x)^4) - x^5*A(x)^10*A(x*A(x)^5) + x^6*A(x)^12*A(x*A(x)^6) + ...

%e (2) 1 = 1/(1 + x*A(x)^2) + 1*x/(1 + x*A(x)^3) + 2*x^2/(1 + x*A(x)^4) + 6*x^3/(1 + x*A(x)^5) + 23*x^4/(1 + x*A(x)^6) + 103*x^5/(1 + x*A(x)^7) + 516*x^6/(1 + x*A(x)^8) + ... + a(n)*x^n/(1 + x*A(x)^(n+2)) + ...

%o (PARI) /* 1 = Sum_{n>=0} (-x)^n * A(x)^(2*n) * A(x*A(x)^n) */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( sum(n=0,#A-1, (-x)^n*Ser(A)^(2*n)*subst(Ser(A),x,x*Ser(A)^n) ),#A-1)); A[n+1]}

%o for(n=0,31,print1(a(n),", "))

%o (PARI) /* 1 = Sum_{n>=0} a(n) * x^n / (1 + x*A(x)^(n+2)) */

%o {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);

%o A[#A] = -polcoeff( sum(n=0,#A-1, A[n+1]*x^n/(1 + x*Ser(A)^(n+2)) ),#A-1)); A[n+1]}

%o for(n=0,31,print1(a(n),", "))

%Y Cf. A352853, A352855, A352856.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Apr 05 2022