A363114 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(4*n-1).

1, 4, 138, 6571, 353935, 20694945, 1276853497, 81834405039, 5395444806588, 363600236084796, 24933767742193052, 1734273108108910743, 122058422998192278797, 8676376795137864622232, 622018188741046650309066, 44922343315319150402783783, 3265215115112327274815579250

Offset

0,2

Links

Table of n, a(n) for n=0..16.

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)^(4*n-1).

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

Example

G.f.: A(x) = 1 + 4*x + 138*x^2 + 6571*x^3 + 353935*x^4 + 20694945*x^5 + 1276853497*x^6 + 81834405039*x^7 + 5395444806588*x^8 + ...

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)^(4*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, x^(4*m^2)/(1 - 2*Ser(A)*x^m)^(4*m+1) ), #A-1)/2); A[n+1]}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A357227, A363112, A363113.

Cf. A361774.

Sequence in context: A155207 A201388 A089666 * A029850 A221675 A340838

Adjacent sequences: A363111 A363112 A363113 * A363115 A363116 A363117

Keyword

nonn

Author

Paul D. Hanna, May 14 2023

Status

approved