A363563 Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n * A(x)^(2*n) / (1 + x^(n+1)*A(x)^3).

1, 2, 11, 84, 738, 7029, 70570, 735401, 7879118, 86249454, 960434270, 10845322135, 123896322956, 1429327711980, 16628329185358, 194858230552674, 2297972689389087, 27252117638208701, 324797817830706494, 3888255542301372866, 46733817274361827340, 563736664663891455990

Offset

0,2

Comments

The g.f. A(x) 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*A(x)^2, q = -x*A(x)^3, and r = x.

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 defined by the following formulas.

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

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

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

Example

G.f.: A(x) = 1 + 2*x + 11*x^2 + 84*x^3 + 738*x^4 + 7029*x^5 + 70570*x^6 + 735401*x^7 + 7879118*x^8 + 86249454*x^9 + 960434270*x^10 + ...
where
1 = 1/(1 + x*A(x)^3) + x*A(x)^2/(1 + x^2*A(x)^3) + x^2*A(x)^4/(1 + x^3*A(x)^3) + x^3*A(x)^6/(1 + x^4*A(x)^3) + x^4*A(x)^8/(1 + x^5*A(x)^3) + ...
also,
1 = 1/(1 - x*A(x)^2) - x*A(x)^3/(1 - x^2*A(x)^2) + x^2*A(x)^6/(1 - x^3*A(x)^2) - x^3*A(x)^9/(1 - x^4*A(x)^2) + x^4*A(x)^12/(1 - x^5*A(x)^2) -+ ...

Prog

(PARI) {a(n, k=3) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-1 + sum(n=0, #A, x^n * Ser(A)^((k-1)*n) / (1 + x^(n+1)*Ser(A)^k ) ), #A)); A[n+1]}
for(n=0, 40, print1(a(n, 3), ", "))

Crossrefs

Cf. A363562, A363564, A340329, A340355.

Sequence in context: A104086 A143140 A191805 * A279202 A086406 A158098

Adjacent sequences: A363560 A363561 A363562 * A363564 A363565 A363566

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2023

Status

approved