A325156 G.f.: A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * sqrt(A(x)))^n / (1 - (-1)^n * x*sqrt(A(x)))^(n+1).

1, 1, 5, 17, 73, 305, 1357, 6113, 28241, 132257, 628181, 3015281, 14611481, 71367505, 351026077, 1737069249, 8642422689, 43204965953, 216918504613, 1093299119057, 5529714637545, 28057603921521, 142778662494957, 728511386126497, 3726298158066673, 19103173934709985, 98140344111905909, 505171574309714993, 2605081666723574969, 13456821571101555345, 69623363954847218493

Offset

0,3

Comments

Compare the g.f. to the following related identities.

(1.a) F(x) = Sum_{n>=0} x^n * (1 + sqrt(F(x)))^n / (1 + x*sqrt(F(x)))^(n+1) holds when G(x) = 1/(1-x).

(1.b) F(x) = Sum_{n>=0} x^n * (1 - sqrt(F(x)))^n / (1 - x*sqrt(F(x)))^(n+1) holds when G(x) = 1/(1-x).

(2.a) G(x) = Sum_{n>=0} x^n * (1 + (-1)^n/sqrt(G(x)))^n / (1 - (-1)^n*x/sqrt(G(x)))^(n+1) holds when G(x) = (1 - x + 4*x^2)/(1 - x)^2.

(2.b) G(x) = Sum_{n>=0} x^n * (1 - (-1)^n/sqrt(G(x)))^n / (1 + (-1)^n*x/sqrt(G(x)))^(n+1) holds when G(x) = (1 - x + 4*x^2)/(1 - x)^2.

Links

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

Formula

G.f. A(x) satisfies:

(1) A(x) = Sum_{n>=0} x^n * (1 + (-1)^n * sqrt(A(x)))^n / (1 - (-1)^n * x*sqrt(A(x)))^(n+1).

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

(3) 0 = 4*x^2*A(x)^2 - (1-x)^2*A(x) + (1-x).

(4) A(x) = ( (1-x)^2 - sqrt( (1-x)*(1 - 3*x - 13*x^2 - x^3) ) )/(8*x^2).

FORMULAS INVOLVING TERMS.

a(n) = 1 (mod 4) for n >= 0.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 17*x^3 + 73*x^4 + 305*x^5 + 1357*x^6 + 6113*x^7 + 28241*x^8 + 132257*x^9 + 628181*x^10 + 3015281*x^11 + 14611481*x^12 + ...
such that A = A(x) satisfies
A(x) = 1/(1 + x*sqrt(A)) + x*(1 + sqrt(A))/(1 - x*sqrt(A))^2 + x^2*(1 - sqrt(A))^2/(1 + x*sqrt(A))^3 + x^3*(1 + sqrt(A))^3/(1 - x*sqrt(A))^4 + x^4*(1 - sqrt(A))^4/(1 + x*sqrt(A))^5 + x^5*(1 + sqrt(A))^5/(1 - x*sqrt(A))^6 + x^6*(1 - sqrt(A))^6/(1 + x*sqrt(A))^7 + x^7*(1 + sqrt(A))^7/(1 - x*sqrt(A))^8 + ...
also,
A(x) = 1/(1 - x*sqrt(A)) + x*(1 - sqrt(A))/(1 + x*sqrt(A))^2 + x^2*(1 + sqrt(A))^2/(1 - x*sqrt(A))^3 + x^3*(1 - sqrt(A))^3/(1 + x*sqrt(A))^4 + x^4*(1 + sqrt(A))^4/(1 - x*sqrt(A))^5 + x^5*(1 - sqrt(A))^5/(1 + x*sqrt(A))^6 + x^6*(1 + sqrt(A))^6/(1 - x*sqrt(A))^7 + x^7*(1 - sqrt(A))^7/(1 + x*sqrt(A))^8 + ...
RELATED SERIES.
sqrt(A(x)) = 1 + 1/2*x + 19/8*x^2 + 117/16*x^3 + 3843/128*x^4 + 30751/256*x^5 + 532887/1024*x^6 + 4692925/2048*x^7 + 341106147/32768*x^8 + ...

Prog

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

Crossrefs

Cf. A325157.

Sequence in context: A149719 A149720 A149721 * A149722 A166228 A362177

Adjacent sequences: A325153 A325154 A325155 * A325157 A325158 A325159

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2019

Status

approved