A363112 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).

1, 1, 6, 51, 470, 4716, 49350, 534115, 5929892, 67175779, 773473709, 9025907984, 106511693025, 1268898400188, 15240421643846, 184348620664449, 2243749948233175, 27459089491691552, 337685454820968084, 4170918486201555250, 51719670553572755173, 643610071084847351183

Offset

0,3

Links

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

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

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

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

Example

G.f.: A(x) = 1 + x + 6*x^2 + 51*x^3 + 470*x^4 + 4716*x^5 + 49350*x^6 + 534115*x^7 + 5929892*x^8 + 67175779*x^9 + 773473709*x^10 + ...

Prog

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

Crossrefs

Cf. A357227, A363113, A363114.

Cf. A361772.

Sequence in context: A292840 A356394 A202754 * A180901 A199685 A225615

Adjacent sequences: A363109 A363110 A363111 * A363113 A363114 A363115

Keyword

nonn

Author

Paul D. Hanna, May 14 2023

Status

approved