The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 (list; graph; refs; listen; history; text; internal format)
 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 Table of n, a(n) for n=0..41. 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 Cf. A001163/A001164, A362114-A362116. Sequence in context: A243361 A037278 A164852 * A033048 A108771 A041308 Adjacent sequences: A362110 A362111 A362112 * A362114 A362115 A362116 KEYWORD sign AUTHOR N. J. A. Sloane, Apr 15 2023 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 18 13:18 EDT 2024. Contains 376000 sequences. (Running on oeis4.)