###The Challenge

Given the number of items, n, in a non-empty, sorted list output the index, i(n), at which its
"Back-To-Front Permutation"
would reside in a list of all permutations if said permutations were sorted lexicographically.

Results may be 0 or 1-based, just say which (that is i, not n).

###The Back-To-Front Permutation

...is the result of building a list of items by repeatedly taking the back (right) then front (left) of a forward sorted (left-to-right) list until all items have been moved to the new list, like so:

Input being consumed     Output being built
----------------------+----------------------
[1,2,3,4,5,6,7]       |   []
[1,2,3,4,5,6]         |   [7]
  [2,3,4,5,6]         |   [7,1]
  [2,3,4,5]           |   [7,1,6]
    [3,4,5]           |   [7,1,6,2]
    [3,4]             |   [7,1,6,2,5]
      [4]             |   [7,1,6,2,5,3]
       []             |   [7,1,6,2,5,3,4]
----------------------+----------------------
                Result:   [7,1,6,2,5,3,4]

###The Permutation Index

If n is 7 (as the above Back-To-Front example) there are 7! = 5040 possible permutations of the (distinct) items.

The first (or zeroth if you prefer) item in the lexicographically sorted list of all those permutations would be [1,2,3,4,5,6,7] itself.
The second item would be [1,2,3,4,5,7,6].
The penultimate item would be [6,7,5,4,3,2,1].
The final item would be [7,6,5,4,3,2,1].

Somewhere in the list is [7,1,6,2,5,3,4] - the Back-To-Front permutation.
In fact it resides at index 4421 (or 4420, 0-based).

The first 100 terms of the (1-based) series of i(n) stating with n=1 are:

[1, 2, 5, 20, 101, 620, 4421, 35900, 326981, 3301820, 36614981, 442386620, 5784634181, 81393657020, 1226280710981, 19696509177020, 335990918918981, 6066382786809020, 115578717622022981, 2317323290554617020, 48773618881154822981, 1075227108896452857020, 24776789629988523782981, 595671612103250915577020, 14915538431227735068422981, 388375922695377900515577020, 10500493527722974260252422981, 294387851083990886241251577020, 8547374142655711068302364422981, 256705485669535347568006115577020, 7966133168508387470157556764422981, 255164703765185142697060455395577020, 8428152915046701352821133945884422981, 286804646124557439494797475697635577020, 10046343320261587490171853861825564422981, 361946983469639629977827594289009635577020, 13401806107756705416338151987291892764422981, 509620811358844406343669072112782398435577020, 19888261269838598952296612667790114958364422981, 796027021978059135393314656928325779313635577020, 32656499591185747972776747396512425885838364422981, 1372349618161694150570365858847999144050545635577020, 59042913445212141486784766209665998363213966364422981, 2599228661343236626556841044804949891956424561635577020, 117022992204136957935406320450852765172427309198364422981, 5385599167607951991914899108349402127789224443761635577020, 253237642343560228651049456045262577841408407945358364422981, 12160677950192512442211239591328112460680077946732401635577020, 596121186084075048430040923729967264426872753432477838364422981, 29817972015629302995182567242334801579950768815528034161635577020, 1521300781271752977229060449226968409483308951201458077838364422981, 79136874389672125594431576407176798565806196489681819746161635577020, 4195746409670353438703582176982222851124537591877131904925838364422981, 226647950929571027033389160506045358232154026979930809227362161635577020, 12469755402728704898931711687060471601348167024469505953048477838364422981, 698528832402134746955113935776664478135149811856698952734398562161635577020, 39828390672475082008725487969655657656845234984369903192450082717838364422981, 2310732940610403489820749422545419026172017083196773021228249831522161635577020, 136372385605079432248118270297843987319730859689490659519593045108637838364422981, 8184614727136310712028222912925520393434441746671755292929684651300962161635577020, 499395599150088488088828589263699706832570087241364247806476254829684637838364422981, 30970577661237849037564293765687064381179710710016867944356691992991422562161635577020, 1951637737743202215078582414596211073163593979517251760161922907619738331037838364422981, 124935294448140961888354806920565269729701922195027940438639971467594965899362161635577020, 8122715297634329704834815499864930982456556629150409552483483162921360809076637838364422981, 536222223779808734298894424747977821661836507759648464980376643706749720339339362161635577020, 35934888694408876553950964671857486605505798806289876128721251856561212716604532637838364422981, 2444100653742421723047039453897314094441893402549077796242989486161660232995578763362161635577020, 168678351774398889649421299427375524997828651490971291597405051437095619521145068660637838364422981, 11809893318195492906423362422261723211461109491055454565957957813190913963268700251019362161635577020, 838668695249666824614744281817664287077123498629740781320472805575397766414810317446260637838364422981, 60395789681636420036909326103457008453700968286067588202502542158402987220806878956757899362161635577020, 4409719671831047920854347812021594101623099731996837427616577550212019116846376438060145780637838364422981, 326378824480107593305098680409232188044060152088938133742995349285199216584125189021190726539362161635577020, 24482761986915290498641378436184801472882183734481184704052899163370643460988742220422624697460637838364422981, 1861011939679134964489290882424961756757512351644848150968435083798473400034549180897307347526539362161635577020, 143322080088606734669581493203883323226982866872563510695813139604263517949121870899167900513721460637838364422981, 11180959098117691096787939665528162905504766712615688479353149686064571807285078895345918312663622539362161635577020, 883437253980179837588356231874303489164303450066956218734514913541773418886216781638015892528346553460637838364422981, 70686019792283622457223177491312228676420353892298796358374930144685265836593932061030928974752467526539362161635577020, 5726440000955084363422511054086796876735936890839327162387490119571704913857298124195153605274993472953460637838364422981, 469637893700329090478715695935318149767077357177154001454773443957172289821041850488811978203204173646406539362161635577020, 38985601803506257421418755484185292421669426050466292273769584084412579273175587484390779961900566697260473460637838364422981, 3275254532761847009577968823645945995578996860191583194845076448298646552018541276645494943006816186458917446539362161635577020, 278435156905293180685369975402415213484477637470382623210256836304261379607777392174394791509334107831816205753460637838364422981, 23948660226767439201080153228038844501800392914958999127628507660415900870134672884615069843391985357739844389446539362161635577020, 2083808638152760278012520365471350750727983345146397213195344003554238214857458501196068353393022808146994627392953460637838364422981, 183398833619245678836784325280074933629492985604252949471226236983335323969170740817904072891411479020269638889458246539362161635577020, 16324556327289215402380134937173544376210173250892288905442294470849835710409338998582008497896189183708810744110298553460637838364422981, 1469391408154472281907142598683652193509359788033796478036774569234135557383656537547410122872987870461908423725867813446539362161635577020, 133730761359685823973259426160811489954077506688872881313704960027919535214176338228137873831877461557289259913042140378553460637838364422981, 12304683293281621431502064899712741587623914209186541475526534622910218175769343180214908250005163885795818227069614613285446539362161635577020, 1144467823788359953327703097406527694627129315367226993710615746590336588945697972034988381266839681418043178062317463477466553460637838364422981, 107592147841885948074037582159380073309559674264815645313786758687454863280472229658194120833316575777142822473140067877053221446539362161635577020, 10222386340397173314525664517235347022088186665852557223898463812546839124314230895213571254552107892786139414391086539473362138553460637838364422981, 981455548530552515895045737024658454136095461985415238220477591025945383684777269092475904782448641089288955324574667766166512421446539362161635577020, 95211304133951567337433380212539040258207718457187560919883999728307800228797098229713403270806624010171995234355103499880901319898553460637838364422981, 9331679144749296178288752362844703433551486045621764102574354777566399269794426700653262755936922495813433855354253356929531746247461446539362161635577020, 923930475294692230638703636199822301473608196598194450583355284174609600662504729388761377005628260366723545352917984225582320362921178553460637838364422981, 92402284968649460451060535220066878189242360067783427018009608611042990392567410879552702599150890025886974375474305774025602890553942821446539362161635577020

(i(0)=i(1)=1, but the challenge itself only deals with non-empty lists)

At the time of posting this sequence did not appear in the OEIS.

Output must only need to work in theory (don't worry about overflowing integers, or running out of resources for example).

This is code-golf, so the shortest answer in bytes wins.

However, don't let code-golf languages dissuade you - good solutions should get upvotes too!