A356503 - OEIS

%I #10 Jan 18 2024 07:33:20

%S 3,82,8856,1319544,227536218,42679033812,8455886664768,

%T 1741107313315440,368888770098828828,79897573332771325074,

%U 17610753240158104125072,3937441977622780631428392,890818276864624495645873656,203562312272030478854160019188,46914726894168080421554447339136

%N G.f. A(x) satisfies: 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).

%H Paul D. Hanna, <a href="/A356503/b356503.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

%F (1) 3 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).

%F (2) 3 = 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 (3) 3 = (-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 a(n) ~ c * d^n / n^(3/2), where d = 256.98186313678886207367060740797984009618789137465012036784040337275... and c = 0.462191657806050531963307936402124127070124175540050112607444087... - _Vaclav Kotesovec_, Mar 19 2023

%e G.f.: A(x) = 3 + 82*x + 8856*x^2 + 1319544*x^3 + 227536218*x^4 + 42679033812*x^5 + 8455886664768*x^6 + 1741107313315440*x^7 + 368888770098828828*x^8 + ...

%e such that A = A(x) satisfies

%e 3 = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...

%t (* 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 -> 3, {r, 1/250}, {s, 2}, WorkingPrecision -> 70] (* _Vaclav Kotesovec_, Jan 18 2024 *)

%o (PARI) {a(n) = my(A=[3],M); for(i=1,n, A = concat(A,0); M = ceil(sqrt(n+1));

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

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A356500, A354248, A356502.

%K nonn

%O 0,1

%A _Paul D. Hanna_, Aug 09 2022