This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A354248 #14 Jan 18 2024 07:28:39
%S A354248 1,2,8,56,446,3892,35904,344560,3404596,34404674,353918256,3693791848,
%T A354248 39016573992,416304888148,4480415851648,48580221470672,
%U A354248 530179054137166,5819280845268420,64197617764575032,711435650735024792,7916252834038848088,88409756229441531072
%N A354248 G.f. A(x) satisfies: 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).
%C A354248 a(n^2) = 2 (mod 4) for n > 0, a(k) = 0 (mod 4) when k > 0 is not square.
%H A354248 Paul D. Hanna, <a href="/A354248/b354248.txt">Table of n, a(n) for n = 0..400</a>
%F A354248 G.f. A(x) satisfies:
%F A354248 (1) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
%F A354248 (2) 1 = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
%F A354248 (3) 1 = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
%F A354248 (4) A(x) = 1 - Sum_{n>=1} (-x)^(n^2) * A(x)^((n-1)^2) * (1 + A(x)^(4*n)).
%F A354248 (5) 1 + x = Sum_{n>=1} A(x)^(n^2) * (-x)^((n-1)^2) * (1 + x^(4*n)).
%F A354248 a(n) ~ c * d^n / n^(3/2), where d = 12.00700721800398362470312030361578928326619844873654254075... and c = 0.185522741203581104495788204096195214315528807577142152... - _Vaclav Kotesovec_, Mar 19 2023
%e A354248 G.f.: A(x) = 1 + 2*x + 8*x^2 + 56*x^3 + 446*x^4 + 3892*x^5 + 35904*x^6 + 344560*x^7 + 3404596*x^8 + 34404674*x^9 + 353918256*x^10 + ...
%e A354248 where
%e A354248 1 = ... + x^16*A(x)^25 - x^9*A(x)^16 + x^4*A(x)^9 - x*A(x)^4 + A(x) - x + x^4*A(x) - x^9*A(x)^4 + x^16*A(x)^9 - x^25*A(x)^16 + ...
%e A354248 equivalently,
%e A354248 A(x) = 1 + x*(1 + A(x)^4) - x^4*A(x)*(1 + A(x)^8) + x^9*A(x)^4*(1 + A(x)^12) - x^16*A(x)^9*(1 + A(x)^16) + x^25*A(x)^16*(1 + A(x)^20) -+ ...
%e A354248 By the Jacobi triple product,
%e A354248 1/A(x) = (1 - x^2*A(x)^2)*(1 - x*A(x)^3)*(1 - x/A(x)) * (1 - x^4*A(x)^4)*(1 - x^3*A(x)^5)*(1 - x^3*A(x)) * (1 - x^6*A(x)^6)*(1 - x^5*A(x)^7)*(1 - x^5*A(x)^3) * (1 - x^8*A(x)^8)*(1 - x^7*A(x)^9)*(1 - x^7*A(x)^5) * ...
%t A354248 (* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], r^2*s^2]) + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / QPochhammer[r^2*s^2, r^2*s^2])) == 0} /. k -> 1, {r, 1/12}, {s, 1}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 18 2024 *)
%o A354248 (PARI) {a(n) = my(A=[1],t); for(i=1,n, A = concat(A,0); t = sqrtint(n+1);
%o A354248 A[#A] = -polcoeff( sum(m=-t,t, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1));A[n+1]}
%o A354248 for(n=0,30,print1(a(n),", "))
%K A354248 nonn
%O A354248 0,2
%A A354248 _Paul D. Hanna_, Jun 18 2022