A363306 - OEIS

%I #10 Jun 23 2023 19:17:21

%S 1,2,5,12,30,79,217,615,1789,5315,16054,49139,152056,474825,1494245,

%T 4733780,15084326,48314504,155459331,502270013,1628784446,5299630868,

%U 17296306669,56607473796,185740962586,610896675767,2013615610286,6650666759129,22007563999578

%N Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n / (1 - (-x)^(n+1)*A(x)).

%C The g.f. of this sequence is motivated by the following identity:

%C Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;

%C here, p = x, q = -x*A(x), and r = -x.

%H Paul D. Hanna, <a href="/A363306/b363306.txt">Table of n, a(n) for n = 0..500</a>

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

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

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

%F (3) 1 = Sum_{n>=0} x^(n^2 + 2*n) * A(x)^n * (1 + x^(2*n+2)*A(x)) / ((1 + (-x)^(n+1))*(1 - (-x)^(n+1)*A(x))).

%e G.f.: A(x) =  1 + 2*x + 5*x^2 + 12*x^3 + 30*x^4 + 79*x^5 + 217*x^6 + 615*x^7 + 1789*x^8 + 5315*x^9 + 16054*x^10 + 49139*x^11 + 152056*x^12 + ...

%e where

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

%e also,

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

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

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

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A340329.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 23 2023