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A363184 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1). 6
1, 4, 18, 88, 452, 2388, 12872, 70520, 391630, 2199816, 12476024, 71341184, 410864744, 2381026908, 13874518912, 81244555896, 477825991140, 2821333839872, 16718050009866, 99385412418648, 592575029005992, 3542752436877800, 21233468105000280, 127555885796445432 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Conjecture: g.f. A(x) == theta_3(x^2) (mod 4); a(n) == 2 (mod 4) if n = 2*k^2 for integer k > 0, and a(n) == 0 (mod 4) if floor(n/2) is nonsquare.
LINKS
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
(1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1).
(2) 4*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 4*A(x)*x^(2*n+1))^(n-1).
(3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^n.
(4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (4*A(x) + x^(2*n-1))^(n-1).
(5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 4*A(x)*x^(2*n+1))^n.
a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 4^(n-2*k) for n >= 0.
EXAMPLE
G.f.: A(x) = 1 + 4*x + 18*x^2 + 88*x^3 + 452*x^4 + 2388*x^5 + 12872*x^6 + 70520*x^7 + 391630*x^8 + 2199816*x^9 + 12476024*x^10 + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (4*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/4); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
Cf. A359670.
Sequence in context: A083325 A050146 A083879 * A081671 A244785 A260650
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 20 2023
STATUS
approved

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Last modified April 15 10:24 EDT 2024. Contains 371681 sequences. (Running on oeis4.)