A369672 - OEIS

%I #16 Feb 03 2024 22:49:32

%S 1,-4,19,-100,569,-3416,21302,-136636,895572,-5971096,40366463,

%T -276036720,1905940182,-13269019988,93040431283,-656472509864,

%U 4657492107245,-33205607204468,237777067846451,-1709374453370956,12332468208675821,-89262196983781332,647988910138661556

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

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

%C Congruences:

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

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

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

%H Paul D. Hanna, <a href="/A369672/b369672.txt">Table of n, a(n) for n = 1..401</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

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

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

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

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

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

%F (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).

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

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

%e 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 + ...

%e where Sum_{n=-oo..+oo} (-1)^n * (x^n - 4*A(x))^n = theta_3(x), and

%e theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + ... + x^(n^2) + ...

%e RELATED SERIES.

%e When we break up the doubly infinite sum into the following parts

%e 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 + ...

%e 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 + ...

%e we see that the sum equals P + N = theta_3(x).

%e SPECIAL VALUES.

%e (V.1) A(exp(-Pi)) = 0.036996905719511834010608252452763733693844226179196126014832...

%e where Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) - 4*A(exp(-Pi)))^n = Pi^(1/4)/gamma(3/4) = 1.0864348112133080...

%e (V.2) A(exp(-2*Pi)) = 0.0018536158947374219405603135305712038712234615914707006019...

%e 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...

%e (V.3) A(exp(-3*Pi)) = 0.0000806734779029429093753810781078431328279003228392603227...

%e 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...

%e (V.4) A(exp(-4*Pi)) = 0.0000034872937107879617892620501277220047637185282553554945...

%e 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...

%e (V.5) A(exp(-5*Pi)) = 0.0000001507016366950287572418174619564191722052174968450159...

%e 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...

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);

%o 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]}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);

%o 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]}

%o for(n=1,30,print1(a(n),", "))

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

%K sign

%O 1,2

%A _Paul D. Hanna_, Feb 03 2024