A363113 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(3*n-1).

1, 2, 30, 621, 14196, 351802, 9179386, 248533626, 6917835992, 196730606200, 5691264122213, 166961281712818, 4955321842136163, 148522859439511133, 4489164688548477495, 136677755757518772050, 4187859771944659634378, 129039023692329781903247, 3995878021838502688832856

Offset

0,2

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)^(3*n-1).

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

Example

G.f.: A(x) = 1 + 2*x + 30*x^2 + 621*x^3 + 14196*x^4 + 351802*x^5 + 9179386*x^6 + 248533626*x^7 + 6917835992*x^8 + 196730606200*x^9 + ...

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)^(3*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, (-1)^(m+1) * x^(3*m^2)/(1 - 2*Ser(A)*x^m)^(3*m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A357227, A363112, A363114.

Cf. A361773.

Sequence in context: A297490 A324957 A282733 * A160694 A278884 A013525

Adjacent sequences: A363110 A363111 A363112 * A363114 A363115 A363116

Keyword

nonn

Author

Paul D. Hanna, May 14 2023

Status

approved