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%I #45 Dec 19 2022 03:44:08
%S 1,2,18,288,6624,198720,7352640,323516160,16499324160,956960801280,
%T 62202452083200,4478576549990400,353807547449241600,
%U 30427449080634777600,2829752764499034316800,282975276449903431680000,30278354580139667189760000,3451732422135922059632640000
%N a(n) = Product_{i=0..n-1} (7*i+2).
%H Harvey P. Dale, <a href="/A084947/b084947.txt">Table of n, a(n) for n = 0..340</a>
%F a(n) = A084942(n)/A000142(n)*A000079(n) = 7^n*Pochhammer(2/7, n) = 7^n*Gamma(n+2/7)/Gamma(2/7).
%F D-finite with recurrence a(0) = 1; a(n) = (7*n - 5)*a(n-1) for n > 0. - _Klaus Brockhaus_, Nov 10 2008
%F G.f.: 1/(1-2*x/(1-7*x/(1-9*x/(1-14*x/(1-16*x/(1-21*x/(1-23*x/(1-28*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 08 2012
%F a(n) = (-5)^n*Sum_{k=0..n} (7/5)^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 From _Ilya Gutkovskiy_, Mar 23 2017: (Start)
%F E.g.f.: 1/(1 - 7*x)^(2/7).
%F a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(3/14)*Gamma(2/7)). (End)
%F Sum_{n>=0} 1/a(n) = 1 + (e/7^5)^(1/7)*(Gamma(2/7) - Gamma(2/7, 1/7)). - _Amiram Eldar_, Dec 19 2022
%p a := n->product(7*i+2,i=0..n-1); [seq(a(j),j=0..30)];
%t Join[{1},FoldList[Times,7*Range[0,15]+2]] (* _Harvey P. Dale_, Nov 27 2015 *)
%t Table[7^n*Pochhammer[2/7, n], {n,0,15}] (* _G. C. Greubel_, Aug 18 2019 *)
%o (Magma) [ 1 ] cat [ &*[ (7*k+2): k in [0..n-1] ]: n in [1..15] ]; // _Klaus Brockhaus_, Nov 10 2008
%o (PARI) vector(20, n, n--; prod(k=0, n-1, 7*k+2)) \\ _G. C. Greubel_, Aug 18 2019
%o (Sage) [product(7*k+2 for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 18 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> 7*k+2) ); # _G. C. Greubel_, Aug 18 2019
%Y Cf. A000079, A000142, A000165, A008544, A001813, A045754, A047055, A047657, A084942, A084947, A084948, A084949, A144739, A144827, A049209, A051188.
%K easy,nonn
%O 0,2
%A Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%E a(15) from _Klaus Brockhaus_, Nov 10 2008
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