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A362113
Truncate Stirling's asymptotic series for 1! after n terms and round to the nearest integer.
3
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 12, 13, -131, -144, 1878, 2047, -31243, -34023, 603493, 656720, -13392786, -14565501, 338472513, 367934625, -9665776360, -10502979551, 309738982467, 336455915833, -11068897604205, -12020303454921, 438669580592210
OFFSET
0,22
COMMENTS
Stirling's series for N! is an asymptotic expansion. It does not converge to N! as more terms are included in the sum.
LINKS
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829. MR1080390 (92b:41049)
FORMULA
In general, we take Stirling's asymptotic series for N! (N >= 1, with N = 1 for the present sequence) and truncate it after n terms. This has the value
sqrt(2*Pi)*N^(N+1/2)*exp(-N)*(Sum_{j = 0..n} c(j)/N^j),
where c(j) = A001163(j)/A001164(j).
We then round this to the nearest integer to get a(n).
MAPLE
h := proc(k) option remember; local j; `if`(k=0, 1, (h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
StirlingAsympt := proc(n) option remember; h(2*n)*2^n*pochhammer(1/2, n) end:
c := n -> StirlingAsympt(n); # # Peter Luschny, Feb 08 2011 (This is A001163(n)/A001164(n)).
S:=proc(k, N) local i; global c; sqrt(2*Pi)*N^(N+1/2)*exp(-N)*add(c(i)/N^i, i=0..k); end;
Digits:=200;
T:=proc(N, M) local k; [seq(round(evalf(S(k, N))), k=0..M)]; end;
T(1, 40);
CROSSREFS
KEYWORD
sign
AUTHOR
N. J. A. Sloane, Apr 15 2023
STATUS
approved