%I #11 Feb 20 2021 14:48:44
%S 1,1,1,1,1,1,1,2,2,1,1,4,6,4,1,1,8,17,17,8,1,1,16,46,63,46,16,1,1,32,
%T 121,216,216,121,32,1,1,64,312,703,907,703,312,64,1,1,128,793,2205,
%U 3538,3538,2205,793,128,1,1,256,1995,6731,13096,16208,13096,6731,1995,256,1,1,512,4984,20139,46640,69476,69476,46640,20139,4984,512,1
%N Triangle of coefficients in g.f. A(x,y) which satisfies: A(x,y) = Sum_{n>=0} x^n/(1 - x*y*A(x,y)^n).
%C The g.f. A(x,y) 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*y, and r = A(x,y).
%F G.f. A(x,y) satisfies:
%F (1) A(x,y) = Sum_{n>=0} x^n/(1 - x*y*A(x,y)^n).
%F (2) A(x,y) = Sum_{n>=0} x^n*y^n/(1 - x*A(x,y)^n).
%F (3) A(x,y) = Sum_{n>=0} x^(2*n) * y^n * A(x)^(n^2) * (1 - x^2*y*A(x)^(2*n)) / ((1 - x*A(x,y)^n)*(1 - x*y*A(x,y)^n)). - _Paul D. Hanna_, Feb 20 2021
%F (4) A(x*y, 1/y) = A(x, y).
%e G.f.: A(x,y) = 1 + (1 + y)*x^1 + (1 + y + y^2)*x^2 + (1 + 2*y + 2*y^2 + y^3)*x^3 + (1 + 4*y + 6*y^2 + 4*y^3 + y^4)*x^4 + (1 + 8*y + 17*y^2 + 17*y^3 + 8*y^4 + y^5)*x^5 + (1 + 16*y + 46*y^2 + 63*y^3 + 46*y^4 + 16*y^5 + y^6)*x^6 + ...
%e where A(x,y) satisfies:
%e A(x,y) = Sum_{n>=0} x^n/(1 - x*y*A(x,y)^n),
%e also
%e A(x,y) = Sum_{n>=0} x^n*y^n/(1 - x*A(x,y)^n).
%e TRIANGLE.
%e This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 2, 1;
%e 1, 4, 6, 4, 1;
%e 1, 8, 17, 17, 8, 1;
%e 1, 16, 46, 63, 46, 16, 1;
%e 1, 32, 121, 216, 216, 121, 32, 1;
%e 1, 64, 312, 703, 907, 703, 312, 64, 1;
%e 1, 128, 793, 2205, 3538, 3538, 2205, 793, 128, 1;
%e 1, 256, 1995, 6731, 13096, 16208, 13096, 6731, 1995, 256, 1;
%e 1, 512, 4984, 20139, 46640, 69476, 69476, 46640, 20139, 4984, 512, 1;
%e 1, 1024, 12397, 59375, 161375, 283599, 340458, 283599, 161375, 59375, 12397, 1024, 1; ...
%o (PARI) {T(n,k) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m/(1 - x*y*A^m +x*O(x^n))) ); polcoeff(polcoeff(H=A,n,x),k,y)}
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%o (PARI) {T(n,k) = my(A=1); for(i=1,n, A = sum(m=0,n, x^m*y^m/(1 - x*A^m +x*O(x^n))) ); polcoeff(polcoeff(H=A,n,x),k,y)}
%o for(n=0,12,for(k=0,n,print1(T(n,k),", "));print(""))
%Y Cf. A340911 (row sums), A340912 (central terms), A340333 (antidiagonal sums).
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Jan 26 2021