On 2016-06-18 17:50, rcs@xmission.com wrote:
Do you have a formula, or a recurrence, for the Poch form?
Ouch! I had this in the clipboard but forgot to paste it: https://en.wikipedia.org/wiki/Stirling%27s_approximation#A_convergent_versio... --rwg
Converting the denominators to factorials/2, starting at 12, I find the numerators are 1, 5, 59, 1218, 38376, ... When these are divided by 12, 60, 360, 2520, 20160, we get the fractions below.
I tried OEIS and Superseeker, no joy.
It would be nice to have a better formula for X!
Rich
--------- Quoting Bill Gosper <billgosper@gmail.com>:
... But with "rising factorials" (Pochhammers) instead of x^n, and weighted row sums of Stirling's Triangle instead of Bernoullis, Out[839]= 1 1 ---------- + ------------------ + 12 (1 + x) 12 (1 + x) (2 + x)
59 --------------------------- + 360 (1 + x) (2 + x) (3 + x)
29 ---------------------------------- + 60 (1 + x) (2 + x) (3 + x) (4 + x)
533 ------------------------------------------- + 280 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x)
1577 --------------------------------------------------- 168 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x) (6 + x)
+ 280361 /
(5040 (1 + x) (2 + x) (3 + x) (4 + x) (5 + x) (6 + x) (7 + x))
Both of these series get added to the usual Stirling approximation ln(x^x/e^x ?(2?x)). The coefficients of this convergent series actually outgrow the Bernoullis, but the Pochhammers grow faster still. ("Pochhammers" is so appealing compared to "rising factorial powers". Too bad that, as usual, Pochhammer didn't invent them.)
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