A369683 - OEIS

%I #11 Feb 05 2024 04:29:03

%S 1,2,4,8,19,40,86,181,383,811,1709,3598,7554,15839,33158,69319,144731,

%T 301813,628727,1308487,2720908,5653743,11740260,24365703,50544832,

%U 104810967,217270721,450287996,933043086,1933125817,4004865745,8296690701,17188106646,35609996584

%N Expansion of g.f. A(x) satisfying Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = theta_3(x).

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

%C a(n+1)/a(n) tends to 2.07474... - _Vaclav Kotesovec_, Feb 05 2024

%H Paul D. Hanna, <a href="/A369683/b369683.txt">Table of n, a(n) for n = 0..361</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.

%F (1) Sum_{n>=0} (-1)^n * x^n * Product_{k=0..n} (x^(2*k+1) + A(x)) = Sum_{n=-oo..+oo} x^(n^2).

%F (2) Sum_{n>=0} (-1)^n * x^(n*(n+1)) / Product_{k=0..n} (1 + x^(2*k+1)*A(x)) = 1 - x * Sum_{n=-oo..+oo} (-1)^n * x^(n^2).

%F (3) theta_3(x) = (x + A(x))/(1 + F(1)), where F(n) = x*(x^(2*n+1) + A(x))/(1 - x*(x^(2*n+1) + A(x)) + F(n+1)), a continued fraction.

%F (4) 1 - x*theta_3(x) = 1/((1 + x*A(x))*(1 + F(1))), where F(n) = x^(2*n) / (1 - x^(2*n) + x^(2*n+1)*A + (1 + x^(2*n+1)*A)*F(n+1)), a continued fraction.

%e G.f.: A(x) = 1 + 2*x + 4*x^2 + 8*x^3 + 19*x^4 + 40*x^5 + 86*x^6 + 181*x^7 + 383*x^8 + 811*x^9 + 1709*x^10 + 3598*x^11 + 7554*x^12 + ...

%e By definition, A = A(x) satisfies the sum of products

%e theta_3(x) = (x + A) - x*(x + A)*(x^3 + A) + x^2*(x + A)*(x^3 + A)*(x^5 + A) - x^3*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A) + x^4*(x + A)*(x^3 + A)*(x^5 + A)*(x^7 + A)*(x^9 + A) -+ ...

%e also, A = A(x) satisfies another sum of products

%e 1 - x*theta_3(x) = 1/(1 + x*A) - x^2/((1 + x*A)*(1 + x^3*A)) + x^6/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)) - x^12/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)*(1 + x^7*A)) + x^20/((1 + x*A)*(1 + x^3*A)*(1 + x^5*A)*(1 + x^7*A)*(1 + x^9*A)) + ...

%e Further, A = A(x) satisfies the continued fraction given by

%e theta_3(x) = (x + A)/(1 + x*(x^3 + A)/(1 - x*(x^3 + A) + x*(x^5 + A)/(1 - x*(x^5 + A) + x*(x^7 + A)/(1 - x*(x^7 + A) + x*(x^9 + A)/(1 - x*(x^9 + A) + x*(x^11 + A)/(1 - ...))))))

%e where theta_3(x) = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + ... + 2*x^(n^2) + ...

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

%o A[#A] = polcoeff( sum(n=-M,M, x^(n^2) ) - sum(n=0,#A, (-1)^n * x^n * prod(k=0,n, x^(2*k+1) + Ser(A)) ), #A-1) ); H=A; A[n+1]}

%o for(n=0,40, print1(a(n),", "))

%Y Cf. A369684, A369682, A369672, A000122 (theta_3).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Feb 04 2024