A363104 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).

1, 6, 44, 348, 2886, 24800, 218888, 1972572, 18075100, 167900506, 1577467760, 14963979584, 143124912880, 1378756186748, 13365212659144, 130274948580864, 1276075285222662, 12554452588117632, 124003727286837484, 1229203475053859456, 12224294019862383720

Offset

0,2

Comments

Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.

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 described as follows.

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

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

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

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

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

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

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

(8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^n)^(n+1).

(9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^n)^n.

(10) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+1))^n.

a(n) = Sum_{k=0..n} A359670(n,k) * 4^k for n >= 0.

Example

G.f.: A(x) = 1 + 6*x + 44*x^2 + 348*x^3 + 2886*x^4 + 24800*x^5 + 218888*x^6 + 1972572*x^7 + 18075100*x^8 + 167900506*x^9 + 1577467760*x^10 + ...

Prog

(PARI) {a(n) = my(A=1, y=4); for(i=1, n,
A = 1/sum(m=-#A, #A, (-1)^m * (x*y*A + x^m + x*O(x^n) )^m ) );
polcoeff( A, n, x)}
for(n=0, 25, print1( a(n), ", "))
(PARI) {a(n) = my(A=[1], y=4); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(-y + sum(n=-#A, #A, (-1)^n * x^n * (y*Ser(A) + x^(n-1))^(n+1) )/(-y), #A-1, x) ); A[n+1]}
for(n=0, 25, print1( a(n), ", "))

Crossrefs

Cf. A359670, A359711, A359712, A359713, A363105.

Cf. A363184.

Sequence in context: A115969 A082412 A108452 * A005591 A052204 A147688

Adjacent sequences: A363101 A363102 A363103 * A363105 A363106 A363107

Keyword

nonn

Author

Paul D. Hanna, May 21 2023

Status

approved