A293130 Constant t defined by: t = Sum_{n>=1} 1 / floor( gamma(n+t)/gamma(t) ).

1, 4, 3, 9, 5, 8, 4, 5, 2, 5, 6, 3, 1, 4, 9, 3, 2, 7, 2, 1, 5, 1, 7, 0, 2, 0, 5, 4, 4, 9, 0, 0, 3, 3, 8, 4, 6, 4, 4, 5, 6, 5, 5, 7, 4, 3, 1, 2, 5, 5, 3, 1, 6, 3, 5, 3, 7, 2, 3, 2, 6, 0, 5, 7, 8, 9, 7, 2, 4, 7, 3, 0, 8, 6, 5, 8, 0, 9, 2, 2, 6, 8, 4, 2, 2, 1, 0, 0, 7, 8, 1, 2, 8, 6, 3, 0, 6, 9, 7, 8, 2, 4, 1, 5, 3, 0, 9, 5, 7, 5, 8, 6, 1, 1, 9, 1, 5, 7, 5, 5, 1, 6, 1, 1, 4, 7, 2, 8, 0, 7, 3, 9, 7, 6, 7, 3, 8, 9, 3, 6, 1, 1, 7, 2, 6, 7, 6, 7, 4, 2, 2, 4, 9, 6, 3, 5, 8, 0, 1, 0, 8, 0, 3, 9, 4, 0, 0, 8, 6, 1, 4, 1, 1, 4, 2, 5, 8, 1, 8, 7, 4, 3, 7, 1, 3, 6, 1, 6, 6, 8, 1, 0, 2, 8, 2, 0, 0, 1, 8, 5, 2

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..2000

Formula

t = Sum_{n>=1} 1 / floor( Product_{k=0..n-1} (k + t) ).

t = Sum_{n>=1} 1/A293131(n), where A293131(n) = floor(Product_{k=0..n-1} (k + t)).

Example

This constant t is defined by
t = 1/[t] + 1/[t*(1+t)] + 1/[t*(1+t)*(2+t)] + 1/[t*(1+t)*(2+t)*(3+t)] + 1/[t*(1+t)*(2+t)*(3+t)*(4+t)] + 1/[t*(1+t)*(2+t)*(3+t)*(4+t)*(5+t)] + 1/[t*(1+t)*(2+t)*(3+t)*(4+t)*(5+t)*(6+t)] + 1/[t*(1+t)*(2+t)*(3+t)*(4+t)*(5+t)*(6+t)*(7+t)] +...
where [x] is the floor function of x.
Explicitly, t is the sum of the infinite series of unit fractions
t = 1 + 1/3 + 1/12 + 1/53 + 1/291 + 1/1878 + 1/13975 + 1/117949 + 1/1113390 + 1/11623335 + 1/132966129 + 1/1654043412 + 1/22229656253 + 1/320987000444 + 1/4955905924999 + 1/81473034355102 + 1/1420855869195491 + 1/26199991898769875 + 1/509316957086997352 + 1/10410226994717110400 + 1/223190941584248205202 + 1/5008311999035018587226 + 1/117392752432115751942460 + 1/2869030095761224977541954 + 1/72986933627698300236793754 + 1/1929744200916184847850410278 + 1/52951379113886857052967930528 + 1/1505915222058143312106047567382 + 1/44333518468215829832469997051113 + 1/1349493882731900596771978592981358 +...+ 1/A293131(n) +...
where A293131(n) = floor( gamma(n+t)/gamma(t) ).
The gamma function of t begins:
gamma(t) = 0.88581292008941981278905841201602316593162655110412781217...
The decimal expansion of the constant begins:
t = 1.43958452563149327215170205449003384644565574312553\
16353723260578972473086580922684221007812863069782\
41530957586119157551611472807397673893611726767422\
49635801080394008614114258187437136166810282001852\
71986524115283147181117613091464099152464344842194\
03130782239819712020783909070772646562174382319601\
87901109174676702574585741493758869423683283302132\
19772471377032093310941373611388876361314271966189\
51687129567401125902522698271243130375515730344144\
89398504298317880132453598772037634155976591780521\
17923774492711461792764326635007336455882638091226\
30796650668192788163602841905506059461656078746236\
24620578796218665453036847516136824206580370036312\
73379175306639573926145224686145601578124507300305\
84188067765705158712515491816705192407489451135262\
86190181616703980708946025822449467960139056972077\
40797366614187428360507507342927211168684236773137\
49294143987080470449032331057452351336014297439184\
50430557156584749123218047693367884635083738677873\
34019674977793547337375843041736088849917290621639\
54389852553420480346353414919523708146738903410993\
01233859364772149707547341029827962425502915272879\
35927631073694435379797551155875907341158582729140\
48357792934899982767160212975485636211452685422411\
97750583668403685526752220714544411739322089041786\
47472784769383100878345814573963489580745215743161\
34101260599898250324408071209287416844546576679828\
79209902086793019530239468625339504507647895589209\
85303626534482276737567070400346537075960151406652\
91201226973295295411721705817522528694811983355579\
30968457719586128867890291158799018076747738874511\
91956170297985333889975995109777626582090431672462\
69746949754661646216920177221895419536721000305683\
43574448536273273114489138972600109548422716852883\
33691388438786126807689424216476053233668428723892\
17238144570510561082073268842749039044087563882981\
11951204498793579405398580185429576405630930517014\
81915539926643164796767912245359092601523055879266\
00302904969244782163649633424417272066067277372356\
12974480645261857901432928896560897591776004010828...

Crossrefs

Cf. A293131.

Sequence in context: A016704 A131896 A114380 * A103825 A073238 A010655

Adjacent sequences: A293127 A293128 A293129 * A293131 A293132 A293133

Keyword

nonn,cons

Author

Paul D. Hanna, Sep 30 2017

Status

approved