login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A293131 a(n) = floor( Product_{k=0..n-1} (k + t) ), where t = Sum_{n>=1} 1/a(n) is given by A293130. 2

%I

%S 1,3,12,53,291,1878,13975,117949,1113390,11623335,132966129,

%T 1654043412,22229656253,320987000444,4955905924999,81473034355102,

%U 1420855869195491,26199991898769875,509316957086997352,10410226994717110400,223190941584248205202,5008311999035018587226,117392752432115751942460,2869030095761224977541954,72986933627698300236793754,1929744200916184847850410278,52951379113886857052967930528,1505915222058143312106047567382

%N a(n) = floor( Product_{k=0..n-1} (k + t) ), where t = Sum_{n>=1} 1/a(n) is given by A293130.

%H Paul D. Hanna, <a href="/A293131/b293131.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = floor( gamma(n + t)/gamma(t) ) for n>=1.

%F a(n) = floor( Sum_{k=1..n} abs( StirlingS1(n, k) ) * t^k ), where StirlingS1(n, k) = A008275(n,k), and t is the constant A293130.

%e The constant t used to define this sequence is defined by

%e 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)] +...

%e where [x] is the floor function of x.

%e Thus, t is the sum of the infinite series

%e 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/a(n) +...

%e Explicitly,The decimal expansion of the constant t begins:

%e t = 1.43958452563149327215170205449003384644565574312553\

%e 16353723260578972473086580922684221007812863069782\

%e 41530957586119157551611472807397673893611726767422\

%e 49635801080394008614114258187437136166810282001852\

%e 71986524115283147181117613091464099152464344842194\

%e 03130782239819712020783909070772646562174382319601\

%e 87901109174676702574585741493758869423683283302132\

%e 19772471377032093310941373611388876361314271966189\

%e 51687129567401125902522698271243130375515730344144\

%e 89398504298317880132453598772037634155976591780521...

%o (PARI) t = 1.4395845256314932721517020544900338464456557431255316353723260578972473

%o {a(n) = floor( prod(k=0,n-1, k + t) )}

%o for(n=1,30, print1(a(n),", "))

%o (PARI) t = 1.4395845256314932721517020544900338464456557431255316353723260578972473

%o {StirlingS1(n, k) = if(n<1, 0, n!*polcoeff(binomial(x, n), k))}

%o {a(n) = floor( sum(k=1,n, abs( StirlingS1(n, k) ) * t^k ) )}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A293130.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 30 2017

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 5 07:06 EDT 2020. Contains 335460 sequences. (Running on oeis4.)