A363107 Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (2*A(x) + x^(n-2))^(n+1).

1, 2, 6, 20, 60, 196, 644, 2128, 7178, 24374, 83496, 288420, 1002272, 3503748, 12311818, 43458316, 154038006, 548018604, 1956263020, 7004845080, 25153186956, 90554989440, 326790211458, 1181910952584, 4283416505940, 15553332981066, 56575492155764, 206136324338908

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

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

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

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

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

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

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 60*x^4 + 196*x^5 + 644*x^6 + 2128*x^7 + 7178*x^8 + 24374*x^9 + 83496*x^10 + 288420*x^11 + 1002272*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1], y=2); 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=2); 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. A359712, A363106, A363108, A363109.

Sequence in context: A000620 A081251 A134293 * A136883 A289173 A057766

Adjacent sequences: A363104 A363105 A363106 * A363108 A363109 A363110

Keyword

nonn

Author

Paul D. Hanna, May 24 2023

Status

approved