A363564 Expansion of g.f. A(x) satisfying 1 = Sum_{n>=0} x^n * A(x)^(3*n) / (1 + x^(n+1)*A(x)^4).

1, 2, 15, 160, 1979, 26633, 378612, 5593669, 85036458, 1321547904, 20901013044, 335307963490, 5443261450865, 89249920538306, 1475910492040246, 24587479259900805, 412252774520658173, 6951447807236206940, 117807212665434783089, 2005490388805271264356

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)^3, q = -x*A(x)^4, 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)^(3*n) / (1 + x^(n+1)*A(x)^4).

(2) 1 = Sum_{n>=0} (-1)^n * x^n * A(x)^(4*n) / (1 - x^(n+1)*A(x)^3).

(3) x = Sum_{n>=1} (-1)^(n-1) * x^(n^2) * A(x)^(7*(n-1)) * (1 + x^(2*n)*A(x)^7) / ((1 - x^n*A(x)^3)*(1 + x^n*A(x)^4)).

Example

G.f.: A(x) = 1 + 2*x + 15*x^2 + 160*x^3 + 1979*x^4 + 26633*x^5 + 378612*x^6 + 5593669*x^7 + 85036458*x^8 + 1321547904*x^9 + 20901013044*x^10 + ...
where
1 = 1/(1 + x*A(x)^4) + x*A(x)^3/(1 + x^2*A(x)^4) + x^2*A(x)^6/(1 + x^3*A(x)^4) + x^3*A(x)^9/(1 + x^4*A(x)^4) + x^4*A(x)^12/(1 + x^5*A(x)^4) + ...
also,
1 = 1/(1 - x*A(x)^3) - x*A(x)^4/(1 - x^2*A(x)^3) + x^2*A(x)^8/(1 - x^3*A(x)^3) - x^3*A(x)^12/(1 - x^4*A(x)^3) + x^4*A(x)^16/(1 - x^5*A(x)^3) -+ ...

Prog

(PARI) {a(n, k=4) = 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, 4), ", "))

Crossrefs

Cf. A363562, A363563, A340329, A340355.

Sequence in context: A139085 A268070 A204679 * A364331 A259608 A317278

Adjacent sequences: A363561 A363562 A363563 * A363565 A363566 A363567

Keyword

nonn

Author

Paul D. Hanna, Jul 19 2023

Status

approved