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A144773
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10-fold factorials: Product_{k=0..n-1} (10*k+1).
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4
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1, 1, 11, 231, 7161, 293601, 14973651, 913392711, 64850882481, 5252921480961, 478015854767451, 48279601331512551, 5359035747797893161, 648443325483545072481, 84946075638344404495011, 11977396665006561033796551, 1808586896415990716103279201, 291182490322974505292627951361
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OFFSET
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0,3
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LINKS
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FORMULA
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a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
E.g.f.: (1 - 10*x)^(-1/10).
a(n) = (-9)^n * Sum_{k = 0..n} (10/9)^k * s(n + 1,n + 1 - k), where s(n, k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 - (10*k+1)*x/( 1 - 10*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 09 2014
a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - Ilya Gutkovskiy, Sep 11 2016
D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - R. J. Mathar, Jan 20 2020
Sum_{n>=0} 1/a(n) = 1 + (e/10^9)^(1/10)*(Gamma(1/10) - Gamma(1/10, 1/10)). - Amiram Eldar, Dec 22 2022
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MAPLE
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G(x):=(1-10*x)^(-1/10): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..14); # Zerinvary Lajos, Apr 03 2009
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MATHEMATICA
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b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* Michael De Vlieger, Sep 14 2016 *)
Join[{1}, FoldList[Times, 10 Range[0, 15]+1]] (* Harvey P. Dale, Oct 24 2022 *)
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PROG
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(PARI) Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ G. C. Greubel, Mar 03 2020
(Magma) R<x>:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // G. C. Greubel, Mar 03 2020
(Sage) [10^n*rising_factorial(1/10, n) for n in (0..15)] # G. C. Greubel, Mar 03 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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