A363109 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-2))^(n+1).

1, 2, 8, 32, 114, 464, 1840, 7424, 30624, 126610, 529832, 2233584, 9471888, 40427152, 173398644, 747197976, 3233336302, 14043404136, 61203859260, 267565075736, 1173030487248, 5156102021680, 22718268675276, 100321210527344, 443919440641296, 1968097221659546

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.

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

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

(3) 4*x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+2))^(n-1).

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

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

(6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (4*A(x) + x^(n-2))^(n-1).

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

(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-1))^n.

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+2))^n.

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 32*x^3 + 114*x^4 + 464*x^5 + 1840*x^6 + 7424*x^7 + 30624*x^8 + 126610*x^9 + 529832*x^10 + 2233584*x^11 + 9471888*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1], y=4); 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=4); 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. A363104, A363106, A363107, A363108.

Sequence in context: A306292 A067897 A145682 * A289451 A230509 A099752

Adjacent sequences: A363106 A363107 A363108 * A363110 A363111 A363112

Keyword

nonn

Author

Paul D. Hanna, May 24 2023

Status

approved