A301831
G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 + x^k*A(x)^k)^k.
Original entry on oeis.org
1, -1, 0, 0, 6, -16, 16, -34, 217, -681, 1343, -3466, 13370, -42380, 109477, -312448, 1040248, -3267138, 9447529, -28367596, 90504001, -283611105, 861087913, -2654231074, 8386506600, -26359974392, 81902319183, -256179313766, 809890745232, -2557697524240, 8046530976599
Offset: 0
G.f. A(x) = 1 - x + 6*x^4 - 16*x^5 + 16*x^6 - 34*x^7 + 217*x^8 - 681*x^9 + 1343*x^10 - 3466*x^11 + ...
log(A(x)) = -x - x^2/2 - x^3/3 + 23*x^4/4 - 51*x^5/5 + 35*x^6/6 - 197*x^7/7 + ... + A281266(n)*x^n/n + ...
A384829
G.f. satisfies A(x) = Sum_{n>=0} x^(n*(n+1)/2) * A(x)^(n*(n+1)).
Original entry on oeis.org
1, 1, 2, 6, 22, 87, 359, 1535, 6758, 30431, 139442, 648001, 3046730, 14467286, 69281190, 334211603, 1622568398, 7921905397, 38871120255, 191586353683, 948083155952, 4708743978840, 23463673225988, 117271827518778, 587744334759630, 2953138645722287, 14872864243128300, 75066312240321173
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 87*x^5 + 359*x^6 + 1535*x^7 + 6758*x^8 + 30431*x^9 + 139442*x^10 + 648001*x^11 + 3046730*x^12 + ...
where A(x) = 1 + x*A(x)^2 + x^3*A(x)^6 + x^6*A(x)^12 + x^10*A(x)^20 + x^15*A(x)^30 + ... + x^(n*(n+1)/2) * A(x)^(n*(n+1)) + ...
-
(* Calculation of constants {d,c}: *) {1/r, 2*(s/Sqrt[Pi*(-5 + 8*r^(7/8)*s^(3/4) * Derivative[0, 0, 2][EllipticTheta][2, 0, Sqrt[r]*s])])} /. FindRoot[{2*r^(1/8)*s^(5/4) == EllipticTheta[2, 0, Sqrt[r]*s], 5*s^(1/4) == 2*r^(3/8) * Derivative[0, 0, 1][EllipticTheta][2, 0, Sqrt[r]*s]}, {r, 1/4}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jun 13 2025 *)
-
{a(n) = my(A = sqrt( (1/x) * serreverse( x*eta(x +x^2*O(x^n))^2/eta(x^2 +x^2*O(x^n))^4 ) ) ); polcoef(A,n)}
for(n=0,30, print1(a(n),", "))