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allocated for Paul DTriangle 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)). Hanna

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, 1, 144, 662, 1277, 1277, 662, 144, 1, 1, 377, 2291, 5719, 7611, 5719, 2291, 377, 1, 1, 987, 7808, 24550, 41593, 41593, 24550, 7808, 987, 1, 1, 2584, 26353, 102299, 214085, 271091

0,8

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

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

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

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

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 + ...

+ ...

where A(x,y) satisfies:

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

also

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

TRIANGLE.

This triangle of coefficients T(n,k) of x^n*y^k in A(x,y) begins

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;

1, 144, 662, 1277, 1277, 662, 144, 1;

1, 377, 2291, 5719, 7611, 5719, 2291, 377, 1;

1, 987, 7808, 24550, 41593, 41593, 24550, 7808, 987, 1;

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

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

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

(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)}

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

(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)}

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

Cf. A340935, A340936, A340910.

allocated

nonn

Paul D. Hanna, Jan 28 2021

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allocated for Paul D. Hanna

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