A340910 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).

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, 121, 216, 216, 121, 32, 1, 1, 64, 312, 703, 907, 703, 312, 64, 1, 1, 128, 793, 2205, 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

Offset

0,8

Comments

The g.f. A(x,y) of this sequence is motivated by the following identity:

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)) ;

here, p = x, q = x*y, and r = A(x,y).

Links

Table of n, a(n) for n=0..77.

Formula

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

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

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

(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

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

Example

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 + ...
where A(x,y) satisfies:
A(x,y) = Sum_{n>=0} x^n/(1 - x*y*A(x,y)^n),
also
A(x,y) = Sum_{n>=0} x^n*y^n/(1 - x*A(x,y)^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, 2, 2, 1;
1, 4, 6, 4, 1;
1, 8, 17, 17, 8, 1;
1, 16, 46, 63, 46, 16, 1;
1, 32, 121, 216, 216, 121, 32, 1;
1, 64, 312, 703, 907, 703, 312, 64, 1;
1, 128, 793, 2205, 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;
1, 1024, 12397, 59375, 161375, 283599, 340458, 283599, 161375, 59375, 12397, 1024, 1; ...

Prog

(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)}
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^m +x*O(x^n))) ); polcoeff(polcoeff(H=A, n, x), k, y)}
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A340911 (row sums), A340912 (central terms), A340333 (antidiagonal sums).

Sequence in context: A191687 A322190 A177254 * A132311 A254414 A199802

Adjacent sequences: A340907 A340908 A340909 * A340911 A340912 A340913

Keyword

nonn,tabl

Author

Paul D. Hanna, Jan 26 2021

Status

approved