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%I #49 Dec 19 2022 03:44:42
%S 1,2,14,168,2856,62832,1696464,54286848,2008613376,84361761792,
%T 3965002804224,206180145819648,11752268311719936,728640635326636032,
%U 48818922566884614144,3514962424815692218368,270652106710808300814336,22193472750286280666775552
%N Quintuple factorial numbers: a(n) = Product_{k=0..n-1} (5*k + 2).
%C Hankel transform is A169621. - _Paul Barry_, Dec 03 2009
%H G. C. Greubel, <a href="/A047055/b047055.txt">Table of n, a(n) for n = 0..350</a>
%F E.g.f. (1-5*x)^(-2/5).
%F a(n) ~ sqrt(2*Pi)/Gamma(2/5)*n^(-1/10)*(5n/e)^n*(1 - (11/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
%F a(n) = A084940(n)/A000142(n)*A000079(n) = 5^n*Pochhammer(2/5, n) = 5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
%F G.f.: 1/(1-2x/(1-5x/(1-7x/(1-10x/(1-12x/(1-15x/(1-17x/(1-20x/(1-22x/(1-25x/(1-.../(1-A047215(n+1)*x/(1-... (continued fraction). - _Paul Barry_, Dec 03 2009
%F a(n) = (-3)^n*Sum_{k=0..n} (5/3)^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 D-finite with recurrence: a(n) +(-5*n+3)*a(n-1) = 0. - _R. J. Mathar_, Dec 03 2012
%F G.f.: 1/G(0) where G(k) = 1 - x*(5*k+2)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 23 2013
%F Sum_{n>=0} 1/a(n) = 1 + (e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). - _Amiram Eldar_, Dec 19 2022
%p a := n->product(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];
%t Table[5^n*Pochhammer[2/5, n], {n,0,20}] (* _G. C. Greubel_, Aug 17 2019 *)
%o (PARI) vector(20, n, n--; prod(k=0,n-1, 5*k+2)) \\ _G. C. Greubel_, Aug 17 2019
%o (Magma) [1] cat [(&*[(5*k+2): k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 17 2019
%o (Sage) [product((5*k+2) for k in (0..n-1)) for n in (0..20)] # _G. C. Greubel_, Aug 17 2019
%o (GAP) List([0..20], n-> Product([0..n-1], k-> (5*k+2) )); # _G. C. Greubel_, Aug 17 2019
%Y Cf. A000165, A008544, A001813, A047657, A084947, A084948, A084949.
%Y Cf. A052562, A008548, A047056, A169621.
%K nonn,easy
%O 0,2
%A Joe Keane (jgk(AT)jgk.org)
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