More comments on A097301:

The sequence B(n)=A(2*n+1)=A097301(n)/A097302(n) is, for n=0..30:

[1/12, -1/60, 2/21, -3, 3360/11, -995040/13, 39916800, -656924748480/17, 

1214047650816000/19, -169382556838010880, 15749593891765493760000/23, 

-4054844479616799289344000, 34017686450062663131463680000, 

-11402327189708082115897599590400000/29, 

189528830020089532044244068728832000000/31, 

-125301437095165913433590881810956288000000, 

3324848840813376377254216201740673351680000000, 

-4160578410821524346413265138571849475899260928000000/37, 

4780430629316161092774167254563413988826782105600000000,

-10344861121462773616229893145148380133812043335614857216000000/41, 

703101417064554835432571392752975595651284650411654381568000000000/43, 

-1288073584847927035953908209903105489803393950945148509827891200000000, 

5744670419206774200105513086562285423246002055193280924028090974208000000000/47, 

-13855862278950288961705793435165043231621368212705546426153969473552384000000000, 

1862302533133202849121804740348378585553697496567169512474285640664146247680000000000, 

-15619697962429757398200186911598890969392842074069697070075581713662771971332505600000000000/53, 

54561934305420662350500046884077627997285542735742874410453774779732250926725988352000000000000, 

-11747787991778276167754816270266769063907873731537764352365986350270123795987587957103001600000000000, 

172608418918694886383615593104585680572735818758858602542043169664257692371671897470312466350080000000000000/59, 

-51140258102373789230726587353864321415119896313875653735989938682131209320778385342816989288202030284800000000000/61, 

275142507731750559746826539943004655568236254306946256870418967587074362346005828844774026209978535116800000000000000]


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The corresponding numerator sequence A097301(n), n=0..30, is:

[1, -1, 2, -3, 3360, -995040, 39916800, -656924748480, 

1214047650816000, -169382556838010880, 15749593891765493760000, 

-4054844479616799289344000, 34017686450062663131463680000, 

-11402327189708082115897599590400000, 

189528830020089532044244068728832000000, 

-125301437095165913433590881810956288000000, 

3324848840813376377254216201740673351680000000, 

-4160578410821524346413265138571849475899260928000000, 

4780430629316161092774167254563413988826782105600000000, 

-10344861121462773616229893145148380133812043335614857216000000, 

703101417064554835432571392752975595651284650411654381568000000000, 

-1288073584847927035953908209903105489803393950945148509827891200000000, 

5744670419206774200105513086562285423246002055193280924028090974208000000000, 

-13855862278950288961705793435165043231621368212705546426153969473552384000000000, 

1862302533133202849121804740348378585553697496567169512474285640664146247680000000000, 

-15619697962429757398200186911598890969392842074069697070075581713662771971332505600000000000, 

54561934305420662350500046884077627997285542735742874410453774779732250926725988352000000000000, 

-11747787991778276167754816270266769063907873731537764352365986350270123795987587957103001600000000000, 

172608418918694886383615593104585680572735818758858602542043169664257692371671897470312466350080000000000000, 

-51140258102373789230726587353864321415119896313875653735989938682131209320778385342816989288202030284800000000000, 

275142507731750559746826539943004655568236254306946256870418967587074362346005828844774026209978535116800000000000000]


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The corresponding denominator sequence A097302 is, for n=0..30:

[12, 60, 21, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1],

and for n=0..50 it is:

[12, 60, 21, 1, 11, 13, 1, 17, 19, 1, 23, 1, 1, 29, 31, 1, 1, 37, 1, 41, 43, 1, 47, 1, 1, 53, 1, 1, 59, 61, 1, 

1, 67, 1, 71, 73, 1, 1, 79, 1, 83, 1, 1, 89, 1, 1, 1, 97, 1, 101, 103]

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The first terms of the series sum(A(2*n+1)*(x^(2*n+1)/(2*n+1))) which appear in the exponent in the derivation of 

Stirling's formula for N! via Euler-Maclaurin summation are, with x=1/N:

1/(12*N) - 1/(360*N^3) + 1/(1260*N^5) + ... 

The coefficients are A(2*n+1)/(2*n+1)! = B(n)/(2*n+1)!.

See the Havil ref. given in A097301 with N in place of his n. 

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