A363108 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^(n+1).

1, 2, 7, 26, 86, 318, 1165, 4312, 16318, 62020, 238165, 921980, 3590145, 14067188, 55399442, 219172028, 870736366, 3472155062, 13892694747, 55759406580, 224427809830, 905659181212, 3663475842865, 14851965523630, 60334690089827, 245572722474460, 1001306332164918

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

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

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

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

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

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

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

(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-1))^n.

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+2))^n.

Example

G.f.: A(x) = 1 + 2*x + 7*x^2 + 26*x^3 + 86*x^4 + 318*x^5 + 1165*x^6 + 4312*x^7 + 16318*x^8 + 62020*x^9 + 238165*x^10 + 921980*x^11 + 3590145*x^12 + ...

Prog

(PARI) {a(n) = my(A=[1], y=3); 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=3); 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. A359713, A363106, A363107, A363109.

Sequence in context: A091145 A261332 A220304 * A300451 A212961 A000697

Adjacent sequences: A363105 A363106 A363107 * A363109 A363110 A363111

Keyword

nonn

Author

Paul D. Hanna, May 24 2023

Status

approved