#Ok to post homework
#Tifany Tong, November 22nd, 2020, HW #20

# 1i) 

# sum(x^(2*n + 1)/(2*n + 1)!, n = 0 .. infinity) = sinh(x) 
# f := taylor(sinh(x)^7/7!, x = 0, 101):
# coeff(f, x, 100)*100! = 0;

# ------------------------------------------------------
# 1ii) 

# f:=exp(x)-1:
# coeff(taylor(add(f^(2*i + 1)/(2*i + 1)!, i = 0 .. 51), x = 0, 101), x, 100)*100! =
# 23792695638183628316167125166708722722054741318989995555688894753342022292474103758734144789060946967960716668504295

# ------------------------------------------------------
# 1iii)
# f := sum(x^n/n!, n = 6 .. infinity) + x^4/24 + x^3/6 + x =
#                               1   5   1  2    
#                f := exp(x) - --- x  - - x  - 1
#                              120      2       

# coeff(taylor(add(f^(2*i + 1)/(2*i + 1)!, i = 0 .. 51), x = 0, 101), x, 100)*100! = 
# 537911979212289319210315022167976910293816540852093547715022508541682212897278245024355264789522538287080267265

# ------------------------------------------------------
# 2i)
# f:=add(x^(2*i+1)/(2*i+1),i=0..101):
# coeff(taylor(f^7/7,x=0,101),x,100)*100! = 0 

# ------------------------------------------------------
# 2ii)
# f := -log(1 - x);
# coeff(taylor(add(f^(2*i + 1)/(2*i + 1)!, i = 0 .. 51), x = 0, 101), x, 100)*100!;
# 4666310772197207634084961942813335024535798413219081073429648194
#   76087999966149578044707319880782591431268489604136118791255926
#   05458432000000000000000000000000


# -------------------------------------------------------
# 2iii)
# f := -log(1 - x) - x^2/2 - x^5/5
# coeff(taylor(add(f^(2*i + 1)/(2*i + 1)!, i = 0 .. 51), x = 0, 101), x, 100)*100!;
# 2317221352396677072412367028030362101063569397191497490198710149
#   83949691345068266709314063224361886921495944993064365978775386
#   08022152025529531964918909222550

# -------------------------------------------------------
# 3) 

# Labeled connected graphs with 100 vertices: 
# f:=log(add(2^(n*(n-1)/2)*x^n/n!, n=0..101)):
# coeff(taylor(f,x=0,101),x,100)*100! = 
# 12545227364841214140113920077308933981143712226232335221646228670073438762733152288570931290464721037132879410785841829335614061038923301089057567492333232742342991703330883362508786575706700912164373533556295772091946436027984956202426848151468051946643334281629530546179260871007606702441804064613292259387215678688204251485113472590740773499865904711614153519223228279103479460149742593313573373922454815379256783977914191368054972202877347560367452385316905184519542107968593433231088205390731399516892353570455670913600968605325975274030857771248499556978365084626972052160824958242001786091884562264466387498757516051526876697136872196315904652955137010457509620268431135862409867267247699476180012636670437828176937847583466292545418672228587485969487224202606879503964238003869150101338752148605336672743521891604530102654762311786819657435002666367293854483842154943967829107914488716935125735393953714388091179550976948529785434582336227066071200707195395804873280783315169923393255351736560235562576092890818834# 4900652899469143373311896574182769657272933821876553379518700197938172155010185719146936187180903250543579433005278984692348282568379624191843385007057047580846112172322370992146748764340302424100769640802926078370838482052406435490036434092467614465282934547622486501083341846045800849550829377015688082924233744178977266360888077466658752803757886543953371541948150838050857811138739570190403175194723845806878286855221273591078542964296886759202533689916185534005248

# Exactly 2 components:
# f := log(add(2^(n*(n - 1)/2)*x^n/n!, n = 0 .. 101)):
# coeff(taylor(f^2/2!, x = 0, 101), x, 100)*100! = 
# 19792878830464097434146485069739463865673187370672887935834214122103592997909477467639145015413302778571558698199938989875722363632867076525500775193313203219625694859620492756307560817786124480904571667305985231399040766859432538780170724152051889531424530851027515639435541861971440089962657458762079240342770560135993633520904674629099681471079341302098763093130534058092714306341148454323169655736417188716167485638876856164406937663195315675056473320830072088996900023615988558834190601024551606490996047256677952803727152702099973371018028710701366236803289614033811563475696670364580819031243080945830466884602280114938472440443402744527681562033913068944126647000531086090506590597863434313244427618995892382119045374120085198882365703112462192132262796290715887651629501613500765319382697459714625232601391113377456170319146465206446792331217136689657731214890373782128820154293953635221308380687242012269185806693761170244502543846560216890121849276503785641853915478872687428718708183133628359872764435735322526# 121955322100244200579269171473679314961687680638065276652910066401714258899512328773662068014373263170965092624777284239415572512416129024249860116530210873861235283964287841875225918264476127609182676153172047392096718128978027994944226801334642136592881198292406724597930004937454732160477796576767954616879358825917759244042876084345101045377871469937328841023216205534951719728974096997883111102605252071058831339598818474925540591534080

# Exactly 100 Components:
# f := log(add(2^binomial(i, 2)*x^i/i!, i = 0 .. 101))^100/100!;
# coeff(taylor(f, x = 0, 101), x, 100)*100! = 1