A363015 - OEIS

%I #8 May 18 2023 10:41:16

%S 1,0,5,13,80,352,1955,10155,56934,316413,1810151,10415443,60776075,

%T 357233548,2118007035,12637190038,75866774437,457815076217,

%U 2775815358337,16900781081347,103294693694125,633491925784696,3897330320229845,24045718580772438,148748241343153325

%N Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} x^n * A(x)^n * (1 + x^n)^(2*n+1).

%H Paul D. Hanna, <a href="/A363015/b363015.txt">Table of n, a(n) for n = 0..300</a>

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

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

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

%e G.f.: A(x) = 1 + 5*x^2 + 13*x^3 + 80*x^4 + 352*x^5 + 1955*x^6 + 10155*x^7 + 56934*x^8 + 316413*x^9 + 1810151*x^10 + ...

%e such that P + Q = 3, where

%e P = 2 + (x*A(x))*(1+x)^3 + (x*A(x))^2*(1+x^2)^5 + (x*A(x))^3*(1+x^3)^7 + (x*A(x))^4*(1+x^4)^9 + (x*A(x))^5*(1+x^5)^11 + (x*A(x))^6*(1+x^6)^13 + ... + (x*A(x))^n*(1+x^n)^(2*n+1) + ...

%e Q = (1/A(x))/(1+x) + (x^2/A(x))^2/(1+x^2)^3 + (x^4/A(x))^3/(1+x^3)^5 + (x^6/A(x))^4/(1+x^4)^7 + (x^8/A(x))^5/(1+x^5)^9 + (x^10/A(x))^6/(1+x^6)^11 + ... + (x^(2*n-2)/A(x))^n/(1+x^n)^(2*n-1) + ...

%e Explicitly,

%e P = 2 + x + 4*x^2 + 9*x^3 + 45*x^4 + 176*x^5 + 948*x^6 + 4621*x^7 + 25196*x^8 + 136099*x^9 + 763772*x^10 + 4319703*x^11 + 24865914*x^12 + ...

%e Q = 1 - x - 4*x^2 - 9*x^3 - 45*x^4 - 176*x^5 - 948*x^6 - 4621*x^7 - 25196*x^8 - 136099*x^9 - 763772*x^10 - 4319703*x^11 - 24865914*x^12 - ...

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

%o A[#A] = polcoeff(-3 + sum(m=-#A,#A, x^m * Ser(A)^m * (1 + x^m +x*O(x^n))^(2*m+1) ), #A-1););A[n+1]}

%o for(n=0,25,print1(a(n),", "))

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 18 2023