A363306 Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n / (1 - (-x)^(n+1)*A(x)).

1, 2, 5, 12, 30, 79, 217, 615, 1789, 5315, 16054, 49139, 152056, 474825, 1494245, 4733780, 15084326, 48314504, 155459331, 502270013, 1628784446, 5299630868, 17296306669, 56607473796, 185740962586, 610896675767, 2013615610286, 6650666759129, 22007563999578

Offset

0,2

Comments

The g.f. of this sequence is motivated by the following identity:

Sum_{n>=0} p^n/(1 - q*r^n) = Sum_{n>=0} q^n/(1 - p*r^n) = Sum_{n>=0} p^n*q^n*r^(n^2)*(1 - p*q*r^(2*n))/((1 - p*r^n)*(1-q*r^n)) ;

here, p = x, q = -x*A(x), and r = -x.

Links

Paul D. Hanna, Table of n, a(n) for n = 0..500

Formula

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

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

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

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

Example

G.f.: A(x) =  1 + 2*x + 5*x^2 + 12*x^3 + 30*x^4 + 79*x^5 + 217*x^6 + 615*x^7 + 1789*x^8 + 5315*x^9 + 16054*x^10 + 49139*x^11 + 152056*x^12 + ...
where
1 = 1/(1 + x*A(x)) + x/(1 - x^2*A(x)) + x^2/(1 + x^3*A(x)) + x^3/(1 - x^4*A(x)) + x^4/(1 + x^5*A(x)) + x^5/(1 - x^6*A(x)) + x^6/(1 + x^7*A(x)) + ...
also,
1 = 1/(1 - x) - x*A(x)/(1 + x^2) + x^2*A(x)^2/(1 - x^3) - x^3*A(x)^3/(1 + x^4) + x^4*A(x)^4/(1 - x^5) - x^5*A(x)^5/(1 + x^6) + x^6*A(x)^6/(1 - x^7) + ...

Prog

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

Crossrefs

Cf. A340329.

Sequence in context: A136704 A120895 A101785 * A003089 A213263 A179544

Adjacent sequences: A363303 A363304 A363305 * A363307 A363308 A363309

Keyword

nonn

Author

Paul D. Hanna, Jun 23 2023

Status

approved