A341233 Denominator of the expected fraction of guests without a napkin in Conway's napkin problem with n guests.

1, 1, 12, 96, 320, 3840, 161280, 516096, 46448640, 185794560, 2270822400, 163499212800, 1821848371200, 51011754393600, 10712468422656000, 9794256843571200, 555007887802368000, 139861987726196736000, 1449478781889675264000, 49059281848573624320000

Offset: 1

Author

Pontus von Brömssen, Feb 07 2021

Keywords

• nonn
• frac

Examples

			0, 0, 1/12, 11/96, 39/320, 473/3840, 19897/161280, 63683/516096, 5731597/46448640
		

Crossrefs

Cf. A248788, A341232 (numerators).

Programs

Python

from sympy import denom, S, factorial
def A341233(n):
  return denom(sum((1-S(2)**(2-k))/factorial(k) for k in range(2,n+1)))

Python

from math import factorial
from fractions import Fraction
def a(n):
  s = sum(Fraction(2**k-4, 2**k*factorial(k)) for k in range(2, n+1))
  return s.denominator
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Feb 07 2021

Formula

A341232(n)/a(n) = Sum_{k=2..n} (1-2^(2-k))/k!.

Lim_{n->oo} A341232(n)/a(n) = (2-sqrt(e))^2 (A248788).