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A293131 a(n) = floor( Product_{k=0..n-1} (k + t) ), where t = Sum_{n>=1} 1/a(n) is given by A293130. 2
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( StirlingS1(n, k) ) * t^k ), where StirlingS1(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 A329056

Adjacent sequences:  A293128 A293129 A293130 * A293132 A293133 A293134

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 30 2017

STATUS

approved

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Last modified June 3 05:29 EDT 2020. Contains 334798 sequences. (Running on oeis4.)