A366026 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + x^k*A(x)^(2*k)).

1, 1, 3, 13, 64, 340, 1903, 11053, 65992, 402508, 2497207, 15709873, 99980007, 642535004, 4164018953, 27181480712, 178559253274, 1179546465168, 7830695860690, 52216823047741, 349584244515573, 2348869478981267, 15833924106623011, 107057382854642578, 725829177205070854

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

A(x) satisfies QPochhammer(-1, x*A(x)^2) = 2*A(x).

a(n) ~ c * d^n / n^(3/2), where d = 7.2188305975020061051473056449576894316519... and c = 0.2182691546096422371919544994005940622002...

Mathematica

nmax = 30; A[_] = 0; Do[A[x_] = Product[1 + x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
(* The constants {d, c}: *) {1/r, 1/(2*Sqrt[Pi*(1/s^2 + 2*r^2*s*Derivative[0, 2][QPochhammer][-1, r*s^2])])} /. FindRoot[{2*s == QPochhammer[-1, r*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s^2] == 1}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120]

Crossrefs

Cf. A171802, A181315.

Sequence in context: A283667 A011272 A367062 * A200719 A065065 A020086

Adjacent sequences: A366023 A366024 A366025 * A366027 A366028 A366029

Keyword

nonn

Author

Vaclav Kotesovec, Sep 26 2023

Status

approved