%I #11 Feb 16 2021 11:37:18
%S 1,1,3,18,126,958,7707,64564,557519,4928784,44398592,406119900,
%T 3762522607,35236156779,333052648607,3173351871750,30448930792460,
%U 293980976342244,2854070686906317,27845872331417000,272896298225616888,2685318944226499424,26521692874280137381
%N G.f. A(x) satisfies: Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^n) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(2*n+1)).
%C Equals row k = 2 of rectangular table A340940.
%F Given g.f. A(x), the following sums are all equal:
%F (1) B(x) = Sum_{n>=0} x^n*A(x)^(2*n)/(1 - x*A(x)^n),
%F (2) B(x) = Sum_{n>=0} x^n*A(x)^n/(1 - x*A(x)^(2*n+1)),
%F (3) B(x) = Sum_{n>=0} x^n/(1 - x*A(x)^(n+2)),
%F (4) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(n^2+2*n) * (1 - x^2*A(x)^(2*n+2)) / ((1 - x*A(x)^n)*(1 - x*A(x)^(n+2))),
%F (5) B(x) = Sum_{n>=0} x^(2*n) * A(x)^(2*n^2+2*n) * (1 + x*A(x)^(2*n+1)) / (1 - x*A(x)^(2*n+1));
%F see the example section for the value of B(x).
%e G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 126*x^4 + 958*x^5 + 7707*x^6 + 64564*x^7 + 557519*x^8 + 4928784*x^9 + 44398592*x^10 + 406119900*x^11 + ...
%e such that the following sums are all equal:
%e B(x) = 1/(1-x) + x*A(x)^2/(1 - x*A(x)) + x^2*A(x)^4/(1 - x*A(x)^2) + x^3*A(x)^6/(1 - x*A(x)^3) + x^4*A(x)^8/(1 - x*A(x)^4) + ...
%e and
%e B(x) = 1/(1-x*A(x)) + x*A(x)/(1 - x*A(x)^3) + x^2*A(x)^2/(1 - x*A(x)^5) + x^3*A(x)^3/(1 - x*A(x)^7) + x^4*A(x)^4/(1 - x*A(x)^9) + ...
%e also
%e B(x) = 1/(1-x*A(x)^2) + x/(1 - x*A(x)^3) + x^2/(1 - x*A(x)^4) + x^3/(1 - x*A(x)^5) + x^4/(1 - x*A(x)^6) + x^5/(1 - x*A(x)^7) + ...
%e where
%e B(x) = 1 + 2*x + 5*x^2 + 18*x^3 + 93*x^4 + 602*x^5 + 4406*x^6 + 34666*x^7 + 286098*x^8 + 2443548*x^9 + 21419500*x^10 + 191631430*x^11 + ...
%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0); H=A; A=concat(A,0);
%o H[#A-1] = -polcoeff( sum(m=0,#A, x^m/(1 - x*Ser(A)^(m+2)) ) - sum(m=0,#A, x^m*Ser(A)^m/(1 - x*Ser(A)^(2*m+1)) ), #A); A=H); A[n+1] }
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A340940, A340941, A340942, A340943, A340895.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 26 2021