A363138 G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) - x^n)^n * (1 - x^n*A(x))^n.

1, 2, 4, 10, 32, 110, 380, 1452, 5444, 21422, 84348, 339498, 1375168, 5638930, 23273316, 96829410, 405119328, 1704806800, 7207449048, 30607173180, 130475372576, 558195492452, 2395675502344, 10312175675736, 44507875822784, 192575428797954, 835133872734696, 3629372408642778

Offset

0,2

Comments

Compare to identity: 0 = Sum_{n=-oo..+oo} x^(2*n) * (y - x^n)^n * (y + x^n)^n, which holds for all y.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:

(1) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) - x^n)^n * (1 - x^n*A(x))^n.

(2) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / ( (A(x) - x^n)^n * (1 - x^n*A(x))^n ).

Example

G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 32*x^4 + 110*x^5 + 380*x^6 + 1452*x^7 + 5444*x^8 + 21422*x^9 + 84348*x^10 + 339498*x^11 + ...
where A = A(x) satisfies
0 = ... - x^12/((A - x^3)^3*(1 - x^3*A)^3) + x^4/((A - x^2)^2*(1 - x^2*A)^2) - 1/((A - x)*(1 - x*A)) + 1 - x^2*(A - x)*(1 - x*A) + x^4*(A - x^2)^2*(1 - x^2*A)^2 - x^6*(A - x^3)^3*(1 - x^3*A)^3 + ...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = -polcoeff(sum(m=-#A, #A, (-1)^m * x^(2*m) * (Ser(A) - x^m)^m*(1 - x^m*Ser(A))^m ), #A-1)); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A354963.

Sequence in context: A296003 A263662 A151400 * A367113 A365516 A336614

Adjacent sequences: A363135 A363136 A363137 * A363139 A363140 A363141

Keyword

nonn

Author

Paul D. Hanna, Jun 12 2023

Status

approved