180831

Ruby


The Stirling-like asymptotic expansion of the hyperfactorial function

A143475(n)/A143476(n) を出力してみた。

def bernoulli(n)
  ary = []
  a = []
  (0..n).each{|i|
    a << 1r / (i + 1)
    i.downto(1){|j| a[j - 1] = j * (a[j - 1] - a[j])}
    ary << a[0] # Bn = a[0]
  }
  ary
end

def A(n)
  a = bernoulli(2 * n + 2)
  ary = [1]
  (1..n).each{|i|
    ary << -1r / (2 * i) * (0..i - 1).inject(0){|s, j| s + a[2 * (i - j) + 2] * ary[j] / ((2 * (i - j) + 1r) * (2 * (i - j) + 2))}
  }
  ary
end

p A(20)

出力結果
[1, (1/720), (-1433/7257600), (1550887/15676416000), (-365236274341/3476402012160000), (31170363588856607/162695614169088000000), (-2626723351027654662151/4919915372473221120000000), (127061942835077684151157039/60219764159072226508800000000), (-5696145248370283185291966600124423/507464726196802564122476544000000000), (254326794362835881966596504823903633657/3288371425755280615513648005120000000000), (-33203124408022060010631772664020406983485604379/49554442037561776763544469977956352000000000000), (5190609778186715223789005366945445257148700136639/728146903409071005505143232329154560000000000000), (-2168800830257199852481630419559118897078575013107378361693/23736657023099351168580622790593057338163200000000000000), (688254812126920339705848920425635906853150484332853559547459091/495621398642314452399963403867583037220847616000000000000000), (-455094637798219508780751141299232276619401642948330067032553732024790741/18429741183082299990226927168424539349021685102673920000000000000000), (1314225236181088962408003668993914937576664551129347651283560067977220345649007/2587535662104754918627860574446805324602644588415418368000000000000000000), (-40137493558162510884237273502402082709643235939576234218825849819173302828302345563043/3354592695427279558259499341168353054428851327573249344143360000000000000000000), (371822015977370080622382857779362209677268088349533503961597215227333604839905242031270387181/1161762542280375456616429811833424029809799791765167712863728435200000000000000000000), (-1016782206914618476145896052640613241494838902632579921164807594651331147940333694004542778848239013/105395097835675661424242512529528227984345037108936014910997443641344000000000000000000000), (1012723610398217335277463645775097080698774145590753907209640008589667619163985343754321082589972196952513/3111263288109145525243638969871673290097865495455791160172644536292474880000000000000000000000), (-18688570877258342166581525345907164051117245537931249574277495598023040337843281852502790748274512478965088968186807/1529437124523418693766279950416474025882276517287009606279360315947964591767552000000000000000000000000)]