A307605 G.f. A(x) satisfies: A(x) = (1 + x) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

1, 1, 2, 5, 12, 20, 48, 81, 169, 305, 580, 1009, 1966, 3338, 6067, 10503, 18730, 31633, 55641, 93151, 160389, 267585, 452762, 747016, 1253644, 2049943, 3390786, 5516227, 9034745, 14572790, 23668066, 37918484, 61042425, 97231826, 155292944, 245774727, 389998116

Offset

0,3

Comments

Weigh transform of A050369.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Formula

G.f.: Product_{k>=1} (1 + x^k)^(k*A074206(k)).

a(n) ~ ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(4 + 2*r)) * exp((2+r)/(1+r) * ((2^(-1-r) - 1) * Gamma(2+r) * zeta(2+r) / zeta'(r))^(1/(2+r)) * n^((1+r)/(2+r))) / (2^(1/50) * sqrt(Pi*(2+r)) * n^((3 + r)/(4 + 2*r))), where r = A107311 is the root of the equation zeta(r) = 2. - Vaclav Kotesovec, Mar 18 2021

Example

G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 20*x^5 + 48*x^6 + 81*x^7 + 169*x^8 + 305*x^9 + ...

Mathematica

terms = 36; A[_] = 1; Do[A[x_] = (1 + x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Crossrefs

Cf. A050369, A074206, A129373, A307604, A307606.

Sequence in context: A356649 A258602 A327065 * A079648 A080838 A244396

Adjacent sequences: A307602 A307603 A307604 * A307606 A307607 A307608

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 18 2019

Status

approved