A363106 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).

1, 2, 5, 14, 36, 98, 271, 752, 2124, 6052, 17375, 50292, 146469, 428992, 1262946, 3734748, 11089366, 33048498, 98819841, 296388284, 891436452, 2688029716, 8124678435, 24611028218, 74702698749, 227177047220, 692084278902, 2111883982538, 6454350205098, 19754469483978

Offset

0,2

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 may be described as follows.

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

(2) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (A(x) + x^(n-1))^n.

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

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

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

(6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (A(x) + x^(n-2))^(n-1).

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

(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-1))^n.

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^n.

Example

G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 36*x^4 + 98*x^5 + 271*x^6 + 752*x^7 + 2124*x^8 + 6052*x^9 + 17375*x^10 + 50292*x^11 + 146469*x^12 + ...

Prog

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

Crossrefs

Cf. A359711, A363107, A363108, A363109, A363140.

Sequence in context: A182890 A102714 A087223 * A005955 A186903 A062197

Adjacent sequences: A363103 A363104 A363105 * A363107 A363108 A363109

Keyword

nonn

Author

Paul D. Hanna, May 24 2023

Status

approved