A268730 - OEIS

%I #24 Sep 08 2022 08:46:15

%S 10,260,10920,633360,46868640,4218177600,447126825600,54549472723200,

%T 7527827235801600,1159285394313446400,197078517033285888000,

%U 36656604168191175168000,7404634041974617383936000,1614210221150466589698048000,377725191749209181989343232000

%N a(n) = Product_{k = 0..n} 2*(8*k + 5).

%H G. C. Greubel, <a href="/A268730/b268730.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = (2^(4*n + 13/4)*Gamma(1/8)*Gamma(n + 13/8))/(sqrt(Pi)*Gamma(1/4)), where Gamma(x) is the gamma function.

%F a(n) = 2*(8*n + 5)*a(n - 1), a(0)=10.

%F Sum_{n>=0} 1/a(n) = (exp(1/16)*(Gamma(5/8) - Gamma(5/8, 1/16)))/(2*sqrt(2)) = 0.10393932939417..., where Gamma(a, x) is the incomplete gamma function.

%F a(n) ~ sqrt(Pi) * 2^(4*n+9/2) * n^(n+9/8) / (Gamma(5/8) * exp(n)). - _Vaclav Kotesovec_, Feb 20 2016

%F G.f.: 10/(1-b(1)x/(1-(b(1)-10)x/(1-b(2)x/(1-(b(2)-10)x/(1-b(3)x/(...)))))), where b(n)=2(5+8n), i.e. 26,42,58,74. - _Benedict W. J. Irwin_, Feb 24 2016

%F a(n) = 2^(n+1)*A147625(n+2). - _R. J. Mathar_, Jun 07 2016

%F E.g.f.: 10/(1 - 16*x)^(13/8). - _Ilya Gutkovskiy_, Jun 07 2016

%e a(0) = (1 + 2 + 3 + 4) = 10;

%e a(1) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) = 260;

%e a(2) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12) = 10920;

%e a(3) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12)*(13 + 14 + 15 + 16) = 633360, etc.

%t FullSimplify[Table[(2^(4 n + 13/4) Gamma[1/8] Gamma[n + 13/8])/(Sqrt[Pi] Gamma[1/4]), {n, 0, 14}]]

%t Table[Product[16 k + 10, {k, 0, n - 1}], {n, 20}] (* _Vincenzo Librandi_, Feb 12 2016 *)

%o (Magma) [&*[(16*k+10): k in [0..n-1]]: n in [1..20]]; // _Vincenzo Librandi_, Feb 12 2016

%o (PARI) x='x+O('x^50); Vec(serlaplace(10/(1 - 16*x)^(13/8))) \\ _G. C. Greubel_, Apr 09 2017

%Y Cf. A000027, A113770, A147630.

%K nonn,easy

%O 0,1

%A _Ilya Gutkovskiy_, Feb 12 2016