A307606 G.f. A(x) satisfies: A(x) = ((1 + x)/(1 - x)) * A(x^2)^2*A(x^3)^3*A(x^4)^4* ... *A(x^k)^k* ...

1, 2, 6, 16, 46, 104, 268, 596, 1406, 3060, 6812, 14356, 30948, 63660, 132328, 267164, 541678, 1072000, 2127052, 4140340, 8060588, 15458948, 29602504, 55990780, 105693252, 197422424, 367793952, 679206200, 1250557768, 2284986580, 4162202864, 7530956532, 13583095710

Offset: 0

Author

Ilya Gutkovskiy, Apr 18 2019

Keywords

• nonn

Comments

Convolution of A307604 and A307605.

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 16*x^3 + 46*x^4 + 104*x^5 + 268*x^6 + 596*x^7 + 1406*x^8 + 3060*x^9 + ...
		

Links

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

Crossrefs

Cf. A050369, A074206, A307604, A307605, A318767.

Programs

Mathematica

terms = 32; A[] = 1; Do[A[x] = (1 + x)/(1 - x) Product[A[x^k]^k, {k, 2, terms}] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]

Formula

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

a(n) ~ ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(50*(2+r))) * exp(12/625 + 2^(1/(2+r) - 1) * (2+r) * ((1 - 2^(2+r)) * Gamma(2+r) * zeta(2+r))^(1/(2+r)) / (zeta'(r)^(1/(2+r)) * (1+r)) * n^((1+r)/(2+r))) / (A^(144/625) * 2^((3 + 2*r)/(50*(2 + r))) * zeta'(r)^(1/(50*(2+r))) * sqrt(Pi*(2+r)) * n^(1/2 + 1/(50*(2+r)))), where r = A107311 is the root of the equation zeta(r)=2 and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 18 2021