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%I #59 Sep 04 2024 11:21:48
%S 1,1,11,231,7161,293601,14973651,913392711,64850882481,5252921480961,
%T 478015854767451,48279601331512551,5359035747797893161,
%U 648443325483545072481,84946075638344404495011,11977396665006561033796551,1808586896415990716103279201,291182490322974505292627951361
%N 10-fold factorials: Product_{k=0..n-1} (10*k+1).
%H G. C. Greubel, <a href="/A144773/b144773.txt">Table of n, a(n) for n = 0..320</a>
%F a(n) = Sum_{k = 0..n} (-10)^(n - k) * A048994(n, k).
%F a(n) = Sum_{k = 0..n} 10^(n - k) * A132393(n, k).
%F E.g.f.: (1 - 10*x)^(-1/10).
%F a(n) = A045757(n), n>0.
%F 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
%F 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
%F a(n) = 10^n * Gamma(n + 1/10) / Gamma(1/10). - _Artur Jasinski_ Aug 23 2016
%F a(n) ~ sqrt(2*Pi)*10^n*n^(n-2/5)/(Gamma(1/10)*exp(n)). - _Ilya Gutkovskiy_, Sep 11 2016
%F D-finite with recurrence: a(n) - (10*n-9)*a(n-1) = 0. - _R. J. Mathar_, Jan 20 2020
%F 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
%p 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
%t b = 10; Table[FullSimplify[b^n*Gamma[n + 1/b]/Gamma[1/b]], {n, 0, 14}] (* _Michael De Vlieger_, Sep 14 2016 *)
%t Join[{1},FoldList[Times,10 Range[0,15]+1]] (* _Harvey P. Dale_, Oct 24 2022 *)
%o (PARI) Vec(serlaplace( (1-10*x)^(-1/10) +O('x^15) )) \\ _G. C. Greubel_, Mar 03 2020
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 15); Coefficients(R!(Laplace( (1-10*x)^(-1/10) ))); // _G. C. Greubel_, Mar 03 2020
%o (Sage) [10^n*rising_factorial(1/10,n) for n in (0..15)] # _G. C. Greubel_, Mar 03 2020
%Y Essentially a duplicate of A045757.
%Y Cf. k-fold factorials: A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A045756 (9), A256268 (combined table).
%Y Cf. A048994, A132393.
%K nonn,easy
%O 0,3
%A _Philippe Deléham_, Sep 21 2008