|
|
A097301
|
|
Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.
|
|
2
|
|
|
1, -1, 2, -3, 3360, -995040, 39916800, -656924748480, 1214047650816000, -169382556838010880, 15749593891765493760000, -4054844479616799289344000, 34017686450062663131463680000, -11402327189708082115897599590400000, 189528830020089532044244068728832000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
The e.g.f. sum( A(2*n+1)*(x^(2*n+1))/(2*n+1)!,n=0..infinity) appears in the Stirling-formula derivation for N! with x=1/N in the exponent and the formula for A(2*n+1):=a(n)/A097302(n), n>=0, is given below. For Stirling's formula see A001163 and A001164.
The rationals A(2*n+1) = B(n):= (2*n)!*Bernoulli(2*(n+1))/(2*(n+1)) = a(n)/A097304(n) with A(2*n):=0 are the logarithmic transform of the rational sequence {A001163(n)/A001164(n)} (inverse of the sequence transform EXP)
|
|
REFERENCES
|
Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 87.
|
|
LINKS
|
|
|
FORMULA
|
a(n)=numerator(B(n)) with B(n):=Bernoulli(2*n+2)*(2*n)!/(2*n+2) and Bernoulli(n)= A027641(n)/A027642(n).
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,frac,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|