More Comments on A097303: Note on Stirling's formula as an example for Euler-Maclaurin summation. See the J. Havil ref. given in A097301. Here we use N in place of his n. For the Euler-Maclaurin formula used in this context take in eq. (9.67), p.455, of the Graham et al. ref. given in A062993 a=1, b=N and f(k)= ln k. ############################################################################ Stirling's formula becomes (no remainder written). N! ~ (N^(N+1/2))*(exp(-N))*sqrt(2*Pi)*sum(r(k)*((1/N)^k)/k!,k=0..infinity) The (signed) rationals r(k):= N(k)/A097303(n), for k=0..32, written in lowest terms, are: [1, 1/12, 1/144, -139/8640, -571/103680, 163879/1741824, 5246819/104509440, -534703531/179159040, -4483131259/2149908480, 432261921612371/1418939596800, 6232523202521089/23838185226240, -25834629665134204969/338068808663040, -1579029138854919086429/20284128519782400, 746590869962651602203151/18723810941337600, 1511513601028097903631961/32097961613721600, -8849272268392873147705987190261/229179445921972224000, -142801712490607530608130701097701/2750153351063666688000, 2355444393109967510921431436000087153/36884409649559764992000, 2346608607351903737647919577082115121863/24343710368709444894720000, -2603072187220373277150999431416562396331667/15374974969711228354560000, -73239727426811935976967471475430268695630993/258299579491148636356608000, 34856851734234401648335623107688675640839679447003/50921917099683588310302720000, 909773124599542506852275229422593983242880452145053/722165369777330888764293120000, -1527335577854677023023224272800947125313629267269390501/376781932057737855007457280000, -183856455668177802003316143799518064719008299958634826921/22606915923464271300447436800000, 2583312098861137963745902036370496943872138148651712093816393/75959237502839951569503387648000, 5180134290822682443757710427952467581918233549140896702364013/70116219233390724525695434752000, -527550309097873396592733540579928993424142983691519876840948418433873/1342024436127098467421810621153280000, -2114866241537081164613223324215572812504648703648482437460602956015127/2300613319075025944151675350548480000, 180394412915538782140015777241228025103785450235726235175126981743099027459/29511315679169298318083559669104640000, 3226140192053936286912811949056082647586604417173687729452086326364208020303641/210710793949268789991116616037407129600000, -10218654456520534088469164280902985100842191028132480093114328858063973003580356809/81565468625523402577206432014480179200000, -327533835569755994270118937132949504563388022431465624898750045057007704351870776569/978785623506280830926477184173762150400000] The values of r(k)/k!, k=0..32, are (maple9, 10 digits, with e-x meaning *10^(-x)): [1., 0.8333333333e-1, 0.3472222222e-2, -0.2681327160e-2, -0.2294720936e-3, 0.7840392217e-3, 0.6972813758e-4, -0.5921664374e-3, -0.5171790908e-4, 0.8394987207e-3, 0.7204895416e-4, -0.1914438499e-2, -0.1625162628e-3, 0.6403362834e-2, 0.5401647679e-3, -0.2952788095e-1, -0.2481743600e-2, .1795401171, 0.1505611304e-1, -1.391801093, -.1165462766, 13.39798546, 1.120804464, -156.8014127, -13.10786302, 2192.555536, 183.1907335, -36101.11929, -3015.077313, 691346.3761, 57721.33636, -15235812.15, -1271735.639] ############################################################################################################## The numerator sequence for the rationals r(k) give N(k), for n=0..33: [1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701, 2355444393109967510921431436000087153, 2346608607351903737647919577082115121863, -2603072187220373277150999431416562396331667, -73239727426811935976967471475430268695630993, 34856851734234401648335623107688675640839679447003, 909773124599542506852275229422593983242880452145053, -1527335577854677023023224272800947125313629267269390501, -183856455668177802003316143799518064719008299958634826921, 2583312098861137963745902036370496943872138148651712093816393, 5180134290822682443757710427952467581918233549140896702364013, -527550309097873396592733540579928993424142983691519876840948418433873, -2114866241537081164613223324215572812504648703648482437460602956015127, 180394412915538782140015777241228025103785450235726235175126981743099027459, 3226140192053936286912811949056082647586604417173687729452086326364208020303641, -10218654456520534088469164280902985100842191028132480093114328858063973003580356809, -327533835569755994270118937132949504563388022431465624898750045057007704351870776569, 230728480231290522008048001606562350241176825393088090041361149344312886801922569309571253] NOTE: This is n o t sequence A001163 even though the depicted first 17 entries coincide. The first discrepency appears for the 33rd entry, namely N(32)=-327533835569755994270118937132949504563388022431465624898750045057007704351870776569, whereas A001163(32)=-11294270192060551526555825418377569122875449049360883617198277415758886356961061261. In fact, N(32)/A001163(32)= 29. ########################################################################################################################## The denominator sequence A097303 is, for n=0..33: [1, 12, 144, 8640, 103680, 1741824, 104509440, 179159040, 2149908480, 1418939596800, 23838185226240, 338068808663040, 20284128519782400, 18723810941337600, 32097961613721600, 229179445921972224000, 2750153351063666688000, 36884409649559764992000, 24343710368709444894720000, 15374974969711228354560000, 258299579491148636356608000, 50921917099683588310302720000, 722165369777330888764293120000, 376781932057737855007457280000, 22606915923464271300447436800000, 75959237502839951569503387648000, 70116219233390724525695434752000, 1342024436127098467421810621153280000, 2300613319075025944151675350548480000, 29511315679169298318083559669104640000, 210710793949268789991116616037407129600000, 81565468625523402577206432014480179200000, 978785623506280830926477184173762150400000, 69404798757718095283877473059594043392000000] NOTE: The first 32 entries coincide with A001164(n)/n!, n>=0, but for the first time for n=32 (33rd entry) one finds: A001164(32)/32!= 8880988975581887254515845120660348492338221235741297614432239616000000000000/32! = 978785623506280830926477184173762150400000/29, in accordance with the mismatch found above for the numerator sequence. ######################################################################################################