A379764 G.f. A(x) satisfies Sum_{n=-oo..+oo} (4^n*A(x)^n - 4*A(x))^(n+1) = 2*theta_4(x) = 2*Sum_{n=-oo..+oo} (-x)^(n^2).

1, 4, 16, 255, 4344, 49104, 543744, 8203012, 130252849, 1857148424, 26419178032, 406394717168, 6385382646008, 98146593920416, 1511699308553216, 23809523580124415, 378539261694956840, 6005705853112822740, 95566360664123435568, 1532797305341786038280, 24704276598971103213152, 398854924231120034019280

Offset: 1

Author

Paul D. Hanna, Jan 13 2025

Keywords

• nonn

Comments

Conjecture: for n > 0, a(n) is odd iff n is a square.

Conjecture: for n > 0, a(n) == (-1)^(n-1) (mod 4) iff n is a square, otherwise a(n) is divisible by 4 if n is nonsquare.

Examples

			G.f.: A(x) = x + 4*x^2 + 16*x^3 + 255*x^4 + 4344*x^5 + 49104*x^6 + 543744*x^7 + 8203012*x^8 + 130252849*x^9 + 1857148424*x^10 + 26419178032*x^11 + 406394717168*x^12 + ...
where Sum_{n=-oo..+oo} ((4*A(x))^n - 4*A(x))^(n+1) = 2 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 + 4*x^64 - 4*x^81 + 4*x^100 + ...
which equals 2*theta_4(x).
RELATED SERIES.
F(x) = Sum_{n=-oo..+oo} (x^n - x)^(n+1) = 2 - x + x^2 - x^3 + 4*x^4 - 3*x^5 - x^6 - x^7 + 13*x^8 - x^9 - 7*x^10 - 10*x^11 + 10*x^12 - x^13 + 22*x^14 - x^15 + 10*x^16 - 23*x^17 - 25*x^18 - x^19 + 43*x^20 - x^21 + 50*x^22 + ... + A378582(n)*x^n + ...
where F(4*A(x)) = 2*theta_4(x).
SPECIAL VALUES.
A(exp(-Pi)) = 0.05428119975241657901052085287161046...
A(exp(-2*Pi)) = 0.0018814995017074835793961430310238023012609799639857...
  if t = 4*A(exp(-2*Pi)), then Sum_{n=-oo..+oo} (t^n - t)^(n+1) = Pi^(1/4)/gamma(3/4) / 2^(1/8).
A(exp(-4*Pi)) = 0.0000034873910031144550359148154357155839216782319232...
  if t = 4*A(exp(-4*Pi)), then Sum_{n=-oo..+oo} (t^n - t)^(n+1) = Pi^(1/4)/gamma(3/4) * (sqrt(2) + 1)^(1/4)/2^(7/16).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..280
• Wikipedia, Theta function.

Crossrefs

Cf. A378582, A002448 (theta_4).

Programs

PARI

{a(n) = my(A=[0,1], Theta4 = 1 + 2*sum(m=1, sqrtint(n+1), (-x)^(m^2) +x^2*O(x^n) ));
for(i=1, n, A=concat(A, 0);
A[#A] = (1/4)*polcoef( sum(m=-#A,#A, (4^m*Ser(A)^m - 4*Ser(A))^(m+1) ) - 2*Theta4, #A-1) ); A[n+1]}
for(n=1, 25, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) 2*theta_4(x) = Sum_{n=-oo..+oo} ((4*A(x))^n - 4*A(x))^(n+1).

(2) 2*theta_4(x) = Sum_{n=-oo..+oo, n<>-1} (4*A(x))^(n*(n-1)) / (1 - (4*A(x))^(n+1))^(n-1).

a(n) ~ c * d^n / n^(3/2), where d = 17.33174551... and c = 0.022472777... - Vaclav Kotesovec, Jan 22 2025