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A144827 Partial products of successive terms of A017029; a(0)=1.

%I A144827 #44 Mar 20 2024 18:36:05
%S A144827 1,4,44,792,19800,633600,24710400,1136678400,60243955200,
%T A144827 3614637312000,242180699904000,17921371792896000,1451631115224576000,
%U A144827 127743538139762688000,12135636123277455360000,1237834884574300446720000,134924002418598748692480000,15651184280557454848327680000
%N A144827 Partial products of successive terms of A017029; a(0)=1.
%H A144827 T. D. Noe, <a href="/A144827/b144827.txt">Table of n, a(n) for n = 0..100</a>
%F A144827 a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
%F A144827 G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F A144827 a(n) = (-3)^n*Sum_{k=0..n} (7/3)^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 A144827 From _Ilya Gutkovskiy_, Mar 23 2017: (Start)
%F A144827 E.g.f.: 1/(1 - 7*x)^(4/7).
%F A144827 a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
%F A144827 a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - _G. C. Greubel_, Feb 22 2022
%F A144827 Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - _Amiram Eldar_, Dec 19 2022
%e A144827 a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...
%t A144827 FoldList[Times,1,Range[4,150,7]] (* _Harvey P. Dale_, Apr 25 2014 *)
%o A144827 (Magma) [ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // _Klaus Brockhaus_, Nov 10 2008
%o A144827 (SageMath) [1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # _G. C. Greubel_, Feb 22 2022
%Y A144827 Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.
%Y A144827 Cf. A049209, A049308, A051188, A084947, A144739, A147585.
%K A144827 nonn
%O A144827 0,2
%A A144827 _Philippe Deléham_, Sep 21 2008
%E A144827 Corrected a(9) by _Vincenzo Librandi_, Jul 14 2011