A384271 G.f. A(x) satisfies -x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

1, 1, 1, 3, 5, 14, 31, 85, 214, 589, 1572, 4385, 12124, 34315, 97006, 277958, 797969, 2310313, 6708311, 19590928, 57386238, 168805975, 497956135, 1473704926, 4372436946, 13007158125, 38779605810, 115872525324, 346897113802, 1040486309806, 3126167631775, 9407946523434, 28355033124335, 85582565615778

Offset: 0

Author

Paul D. Hanna, May 24 2025

Keywords

• nonn

Comments

The g.f. utilizes the Jacobi triple product identity: Product_{n>=1} (1 - x^n/a)*(1 - x^(n-1)*a)*(1-x^n) = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)/2) * a^n.

Examples

			G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 5*x^4 + 14*x^5 + 31*x^6 + 85*x^7 + 214*x^8 + 589*x^9 + 1572*x^10 + 4385*x^11 + 12124*x^12 + ...
RELATED SERIES.
1/A(x) = 1 - x - 2*x^3 - 7*x^5 - 4*x^6 - 33*x^7 - 43*x^8 - 190*x^9 - 363*x^10 - 1265*x^11 - 2967*x^12 - 9313*x^13 - 24254*x^14 + ...
By definition of g.f. A(x),
-x = (1 - x/A(x))*(1 - A(x))*(1 + x) * (1 - x^2/A(x))*(1 - x*A(x))*(1 + x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1 + x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1 + x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1 + x^5) * (1 - x^6/A(x))*(1 - x^5*A(x))*(1 + x^6) * ...
also,
-x*theta_4(x) = (1 - x/A(x))*(1 - A(x))*(1 - x) * (1 - x^2/A(x))*(1 - x*A(x))*(1 - x^2) * (1 - x^3/A(x))*(1 - x^2*A(x))*(1 - x^3) * (1 - x^4/A(x))*(1 - x^3*A(x))*(1 - x^4) * (1 - x^5/A(x))*(1 - x^4*A(x))*(1 - x^5) * (1 - x^6/A(x))*(1 - x^5*A(x))*(1 - x^6) * ...
where Jacobi's theta_4(x) begins
theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 - 2*x^25 + 2*x^36 - 2*x^49 +- ... + (-1)^n*2*x^(n^2) + ...
SPECIFIC VALUES.
A(exp(-Pi)) = 1.0453432348429282081117266580603161092013621219944501002...
  where Sum_{n=-oo..+oo} (-1)^n * exp(-Pi*n*(n-1)/2) * A(exp(-Pi))^n = -exp(-Pi) * (Pi/2)^(1/4) / gamma(3/4) = -0.03947933420376592813...
A(-exp(-Pi)) = 0.958426933091195985748561440955710208995111661258536170...
  where Sum_{n=-oo..+oo} (-1)^(n*(n+1)/2) * exp(-Pi*n*(n-1)/2) * A(-exp(-Pi))^n = exp(-Pi) * Pi^(1/4) / gamma(3/4) = 0.04694910513068872743...
A(t) = 2 at t = 0.31637346425553975249950084871655397381494910538235011...
A(t) = 7/4 at t = 0.306394408393287726599555143524924576884132332626742...
A(t) = 5/3 at t = 0.298403642258683011765026172638519982558475148161098...
A(t) = 3/2 at t = 0.271351341798078045586394278854619398226629821704419...
A(t) = 4/3 at t = 0.222121640630627872529588897705597278294416500502588...
A(t) = 5/4 at t = 0.185212111226798258067304643213927542314746099159395...
A(1/4) = 1.415936196810577322060687637240440296052642753467849...
A(1/5) = 1.280767471524969264389815996502959550941291484191129...
A(1/6) = 1.215194363106761985540779108431983083763494550900814...
A(1/7) = 1.175354795171732738951963612236910785381681269370988...
A(1/8) = 1.148310502549415307985734864677154956069415167149368...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..630

Crossrefs

Cf. A356499, A384272, A384273.

Programs

Mathematica

(* Calculation of constants {d, c}: *) {1/r, -s*Log[r] * Sqrt[((s-1)*(-2*r*(s-1) * QPochhammer[s, r] * Derivative[0, 1][QPochhammer][-1, r] + s*QPochhammer[-1, r]^2*QPochhammer[s, r]^2 * Derivative[0, 1][QPochhammer][1/s, r] + 2*(s-1) * QPochhammer[-1, r] * (QPochhammer[s, r] - r*Derivative[0, 1][QPochhammer][s, r]))) / (QPochhammer[-1, r] * QPochhammer[s, r] * (-s*Log[r]^2 + (s-1)^2 * QPolyGamma[1, -Log[s]/Log[r], r] + (s-1)^2 * QPolyGamma[1, Log[s]/Log[r], r]))] / (2*Sqrt[Pi])} /. FindRoot[{s * QPochhammer[-1, r] * QPochhammer[1/s, r] * QPochhammer[s, r] == -2*r*(s-1), Log[r]/(s-1) == QPolyGamma[0, Log[1/s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]}, {r, 1/3}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, May 25 2025 *)

PARI

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas where theta_4(x) is a Jacobi elliptic function.

(1) -x = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 + x^n).

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

(3) -x*theta_4(x) = Product_{n>=1} (1 - x^n/A(x)) * (1 - x^(n-1)*A(x)) * (1 - x^n).

(4) x*theta_4(x)/A(x) = Product_{n>=1} (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^n).

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

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

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

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

a(n) ~ c * d^n / n^(3/2), where d = 3.15858040658396206484741188... and c = 0.5457701830227905480303... - Vaclav Kotesovec, May 25 2025