A369672 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x).

1, -4, 19, -100, 569, -3416, 21302, -136636, 895572, -5971096, 40366463, -276036720, 1905940182, -13269019988, 93040431283, -656472509864, 4657492107245, -33205607204468, 237777067846451, -1709374453370956, 12332468208675821, -89262196983781332, 647988910138661556

Offset

1,2

Comments

Note: theta_3(x) = Sum_{n=-oo..+oo} x^(n^2) - see A000122.

Congruences:

(C.1) a(2*n) == 0 (mod 4) for n >= 1.

(C.2) a(n) == A369671(n) (mod 4) for n >= 1.

(C.3) a(2*n)/4 == -A369671(2*n)/4 (mod 4) for n >= 1.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..401

Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Formula

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

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

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

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

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

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

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

a(n) ~ c * (-1)^(n+1) * d^n / n^(3/2), where d = 7.7471235933114571108403244715948697607... and c = 0.26329435412874059034137968338302672... - Vaclav Kotesovec, Feb 03 2024

Example

G.f.: A(x) = x - 4*x^2 + 19*x^3 - 100*x^4 + 569*x^5 - 3416*x^6 + 21302*x^7 - 136636*x^8 + 895572*x^9 - 5971096*x^10 + 40366463*x^11 - 276036720*x^12 + ...
where Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x), and
theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + ... + x^(n^2) + ...
RELATED SERIES.
When we break up the doubly infinite sum into the following parts
P = Sum_{n>=0} (-1)^n * (x^n - 4*A(x))^n = 1 + 3*x + 4*x^3 - 15*x^4 + 92*x^5 - 528*x^6 + 3196*x^7 - 20032*x^8 + 128819*x^9 - 845312*x^10 + 5638568*x^11 - 38122176*x^12 + ...
N = Sum_{n>=1} (-1)^n * x^(n^2) / (1 - 4*x^n*A(x))^n = -x - 4*x^3 + 17*x^4 - 92*x^5 + 528*x^6 - 3196*x^7 + 20032*x^8 - 128817*x^9 + 845312*x^10 - 5638568*x^11 + 38122176*x^12 + ...
we see that the sum equals P + N = theta_3(x).
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.036996905719511834010608252452763733693844226179196126014832...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) - 4*A(exp(-Pi)))^n = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...
(V.2) A(exp(-2*Pi)) = 0.0018536158947374219405603135305712038712234615914707006019...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) - 4*A(exp(-2*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(2 + sqrt(2))/2 = 1.0037348854877390...
(V.3) A(exp(-3*Pi)) = 0.0000806734779029429093753810781078431328279003228392603227...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-3*n*Pi) - 4*A(exp(-3*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt(1 + sqrt(3))/(108)^(1/8) = 1.000161399035140...
(V.4) A(exp(-4*Pi)) = 0.0000034872937107879617892620501277220047637185282553554945...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-4*n*Pi) - 4*A(exp(-4*Pi)))^n = Pi^(1/4)/gamma(3/4) * (2 + 8^(1/4))/4 = 1.000161399035140...
(V.5) A(exp(-5*Pi)) = 0.0000001507016366950287572418174619564191722052174968450159...
where Sum_{n=-oo..+oo} (-1)^n * (exp(-5*n*Pi) - 4*A(exp(-5*Pi)))^n = Pi^(1/4)/gamma(3/4) * sqrt((2 + sqrt(5))/5) = 1.0000003014034550...

Prog

(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M, M, x^(n^2)) - sum(n=-#A, #A, (-1)^n * (x^n - 4*x*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); M=sqrtint(#A+4);
A[#A] = polcoeff( (sum(n=-M, M, x^(n^2)) - sum(n=-#A, #A, (-1)^n * x^(n^2)/(1 - 4*x^(n+1)*Ser(A))^n) )/4, #A); ); A[n]}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A369671 (dual), A000122 (theta_3), A355868.

Sequence in context: A083882 A007564 A218185 * A086624 A307121 A261490

Adjacent sequences: A369669 A369670 A369671 * A369673 A369674 A369675

Keyword

sign

Author

Paul D. Hanna, Feb 03 2024

Status

approved