A362113 Truncate Stirling's asymptotic series for 1! after n terms and round to the nearest integer.

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

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