A370034 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).

1, 4, 15, 53, 185, 711, 3270, 17297, 95108, 511258, 2653139, 13479835, 68633758, 356913516, 1906525759, 10388550830, 57084621325, 313692565172, 1719365476703, 9416232699651, 51699722653269, 285294478988749, 1583233662850172, 8826549215612727, 49354550054780111, 276444281747417079

Offset: 1

Author

Paul D. Hanna, Feb 10 2024

Keywords

• nonn

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			G.f.: A(x) = x + 4*x^2 + 15*x^3 + 53*x^4 + 185*x^5 + 711*x^6 + 3270*x^7 + 17297*x^8 + 95108*x^9 + 511258*x^10 + 2653139*x^11 + 13479835*x^12 + ...
where
Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = 1 - 2*x - 2*x^4 - 2*x^9 - 2*x^16 - 2*x^25 - 2*x^36 - 2*x^49 - ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05211271680112049721451382589099198923178830298930738503...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A)^n = 2 - Pi^(1/4)/gamma(3/4) = 0.913565188786691985...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881490436109324727231096204943046774873234177072692211...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A)^n = 2 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.99626511451226...
(V.3) Let A = A(-exp(-Pi)) = -0.03679381086518350821622244996144281973183248006035375080...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 4*A)^n = 2 - (Pi/2)^(1/4)/gamma(3/4) = 1.08642086184388317...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001853590408074327278987912837104527635895010708605840824...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 4*A)^n = 2 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.003734885439...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..401
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370041, A370030, A370031, A355868, A370033, A370035, A370036, A370037, A370038, A370039, A370043.

Cf. A369671.

Programs

PARI

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

Formula

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

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

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

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

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

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

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