A293131 - OEIS

A293131

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

1, 3, 12, 53, 291, 1878, 13975, 117949, 1113390, 11623335, 132966129, 1654043412, 22229656253, 320987000444, 4955905924999, 81473034355102, 1420855869195491, 26199991898769875, 509316957086997352, 10410226994717110400, 223190941584248205202, 5008311999035018587226, 117392752432115751942460, 2869030095761224977541954, 72986933627698300236793754, 1929744200916184847850410278, 52951379113886857052967930528, 1505915222058143312106047567382

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..500

FORMULA

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

a(n) = floor( Sum_{k=1..n} abs( Stirling1(n, k) ) * t^k ), where Stirling1(n, k) = A008275(n,k), and t is the constant A293130.

EXAMPLE

The constant t used to define this sequence 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)] +...

where [x] is the floor function of x.

Thus, t is the sum of the infinite series

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) +...

Explicitly,The decimal expansion of the constant t begins:

t = 1.43958452563149327215170205449003384644565574312553\

16353723260578972473086580922684221007812863069782\

41530957586119157551611472807397673893611726767422\

49635801080394008614114258187437136166810282001852\

71986524115283147181117613091464099152464344842194\

03130782239819712020783909070772646562174382319601\

87901109174676702574585741493758869423683283302132\

19772471377032093310941373611388876361314271966189\

51687129567401125902522698271243130375515730344144\

89398504298317880132453598772037634155976591780521...

PROG

(PARI) t = 1.4395845256314932721517020544900338464456557431255316353723260578972473

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

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

(PARI) t = 1.4395845256314932721517020544900338464456557431255316353723260578972473

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

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

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

CROSSREFS

Cf. A293130.

Sequence in context: A302188 A060460 A306525 * A120983 A124810 A370821

Adjacent sequences: A293128 A293129 A293130 * A293132 A293133 A293134

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 30 2017

STATUS

approved