A097301 Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.

1, -1, 2, -3, 3360, -995040, 39916800, -656924748480, 1214047650816000, -169382556838010880, 15749593891765493760000, -4054844479616799289344000, 34017686450062663131463680000, -11402327189708082115897599590400000, 189528830020089532044244068728832000000

Offset: 0

Author

Wolfdieter Lang, Aug 13 2004

Keywords

• sign
• frac
• easy

Comments

Denominators are given in A097302.

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

• W. Lang, More terms and comments.
• N. J. A. Sloane, Transforms

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).