A370034 - OEIS

%I #14 Feb 16 2025 08:34:06

%S 1,4,15,53,185,711,3270,17297,95108,511258,2653139,13479835,68633758,

%T 356913516,1906525759,10388550830,57084621325,313692565172,

%U 1719365476703,9416232699651,51699722653269,285294478988749,1583233662850172,8826549215612727,49354550054780111,276444281747417079

%N 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).

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

%H Paul D. Hanna, <a href="/A370034/b370034.txt">Table of n, a(n) for n = 1..401</a>

%H Eric Weisstein's World of Mathematics, <a href="https://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} (x^n - 4*A(x))^n = 1 - 2*Sum_{n>=1} x^(n^2).

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

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

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

%e 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 + ...

%e where

%e 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 - ...

%e SPECIAL VALUES.

%e (V.1) Let A = A(exp(-Pi)) = 0.05211271680112049721451382589099198923178830298930738503...

%e then Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A)^n = 2 - Pi^(1/4)/gamma(3/4) = 0.913565188786691985...

%e (V.2) Let A = A(exp(-2*Pi)) = 0.001881490436109324727231096204943046774873234177072692211...

%e 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...

%e (V.3) Let A = A(-exp(-Pi)) = -0.03679381086518350821622244996144281973183248006035375080...

%e then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 4*A)^n = 2 - (Pi/2)^(1/4)/gamma(3/4) = 1.08642086184388317...

%e (V.4) Let A = A(-exp(-2*Pi)) = -0.001853590408074327278987912837104527635895010708605840824...

%e 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...

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

%o 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]}

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

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

%Y Cf. A369671.

%K nonn,changed

%O 1,2

%A _Paul D. Hanna_, Feb 10 2024