A340934 - OEIS

%I #6 Jan 28 2021 21:40:55

%S 1,1,1,1,1,1,1,3,3,1,1,8,13,8,1,1,21,51,51,21,1,1,55,187,268,187,55,1,

%T 1,144,662,1277,1277,662,144,1,1,377,2291,5719,7611,5719,2291,377,1,1,

%U 987,7808,24550,41593,41593,24550,7808,987,1,1,2584,26353,102299,214085,271091

%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)^(2*n)).

%F G.f. A(x,y) satisfies:

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

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

%F (3) A(x*y, 1/y) = A(x, y).

%e G.f.: A(x,y) = 1 + (1 + y)*x + (1 + y + y^2)*x^2 + (1 + 3*y + 3*y^2 + y^3)*x^3 + (1 + 8*y + 13*y^2 + 8*y^3 + y^4)*x^4 + (1 + 21*y + 51*y^2 + 51*y^3 + 21*y^4 + y^5)*x^5 + (1 + 55*y + 187*y^2 + 268*y^3 + 187*y^4 + 55*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)^(2*n)),

%e also

%e A(x,y) = Sum_{n>=0} x^n*y^n/(1 - x*A(x,y)^(2*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, 3, 3, 1;

%e 1, 8, 13, 8, 1;

%e 1, 21, 51, 51, 21, 1;

%e 1, 55, 187, 268, 187, 55, 1;

%e 1, 144, 662, 1277, 1277, 662, 144, 1;

%e 1, 377, 2291, 5719, 7611, 5719, 2291, 377, 1;

%e 1, 987, 7808, 24550, 41593, 41593, 24550, 7808, 987, 1;

%e 1, 2584, 26353, 102299, 214085, 271091, 214085, 102299, 26353, 2584, 1;

%e 1, 6765, 88477, 417543, 1055893, 1638186, 1638186, 1055893, 417543, 88477, 6765, 1;

%e 1, 17711, 296546, 1680731, 5050791, 9377929, 11458077, 9377929, 5050791, 1680731, 296546, 17711, 1; ...

%o (PARI) {T(n, k) = my(A=1); for(i=1, n, A = sum(m=0, n, x^m/(1 - x*y*A^(2*m) +x*O(x^n))) ); polcoeff(polcoeff(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^(2*m) +x*O(x^n))) ); polcoeff(polcoeff(A, n, x), k, y)}

%o for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

%Y Cf. A340935, A340936, A340910.

%K nonn

%O 0,8

%A _Paul D. Hanna_, Jan 28 2021