This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049210 #63 Dec 20 2022 03:49:59
%S A049210 1,7,105,2415,74865,2919735,137227545,7547514975,475493443425,
%T A049210 33760034483175,2667042724170825,232032717002861775,
%U A049210 22043108115271868625,2270440135873002468375,252018855081903273989625,29990243754746489604765375,3808760956852804179805202625
%N A049210 a(n) = -Product_{k=0..n} (8*k-1); octo-factorial numbers.
%H A049210 Seiichi Manyama, <a href="/A049210/b049210.txt">Table of n, a(n) for n = 0..333</a>
%H A049210 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A049210 a(n) = 7*A034975(n) = (8*n-1)(!^8), n >= 1, a(0) = 1.
%F A049210 G.f.: 1/(1-7*x/(1-8*x/(1-15*x/(1-16*x/(1-23*x/(1-24*x/(1-31*x/(1-32*x/(1-... (continued fraction). - _Philippe Deléham_, Jan 07 2012
%F A049210 a(n) = (-1)^n*Sum_{k=0..n} 8^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 A049210 G.f.: ( 1 - 1/Q(0) )/x where Q(k) = 1 - x*(8*k-1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F A049210 a(n) = 8^n*Gamma(n+7/8)/Gamma(7/8). - _R. J. Mathar_, Mar 20 2013
%F A049210 E.g.f: (1-8*x)^(-7/8). - _Vaclav Kotesovec_, Jan 28 2015
%F A049210 G.f.: 1/(1-7*x-56*x^2/(1-23*x-240*x^2/(1-39*x-552*x^2/(1-55*x-992*x^2/(1-71*x-1560*x^2/(1-... )))))) (Jacobi continued fraction). - _Nikolaos Pantelidis_, Dec 09 2020
%F A049210 G.f.: 1/G(0) where G(k) = 1 - (16*k+7)*x - 8*(k+1)*(8*k+7)*x^2/G(k+1); (continued fraction). - _Nikolaos Pantelidis_, Dec 19 2020
%F A049210 Sum_{n>=0} 1/a(n) = 1 + (e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). - _Amiram Eldar_, Dec 20 2022
%t A049210 FoldList[Times,1,8*Range[20]-1] (* _Harvey P. Dale_, Aug 03 2014 *)
%t A049210 CoefficientList[Series[(1-8*x)^(-7/8),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2015 *)
%o A049210 (PARI) a(n) = -prod(k=0, n, 8*k-1); \\ _Michel Marcus_, Jan 08 2015
%o A049210 (Sage) m=8; [m^n*rising_factorial((m-1)/m, n) for n in (0..30)] # _G. C. Greubel_, Feb 16 2022
%o A049210 (Magma) m:=8; [Round(m^n*Gamma(n +(m-1)/m)/Gamma((m-1)/m)): n in [0..30]]; // _G. C. Greubel_, Feb 16 2022
%Y A049210 Cf. A034975, A048994, A051187.
%Y A049210 Sequences of the form m^n*Pochhammer((m-1)/m, n): A000007 (m=1), A001147 (m=2), A008544 (m=3), A008545 (m=4), A008546 (m=5), A008543 (m=6), A049209 (m=7), this sequence (m=8), A049211 (m=9), A049212 (m=10), A254322 (m=11), A346896 (m=12).
%K A049210 easy,nonn
%O A049210 0,2
%A A049210 _Wolfdieter Lang_