OFFSET
0,1
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..300
FORMULA
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.
a(n) = 2*(8*n + 5)*a(n - 1), a(0)=10.
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.
a(n) ~ sqrt(Pi) * 2^(4*n+9/2) * n^(n+9/8) / (Gamma(5/8) * exp(n)). - Vaclav Kotesovec, Feb 20 2016
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
a(n) = 2^(n+1)*A147625(n+2). - R. J. Mathar, Jun 07 2016
E.g.f.: 10/(1 - 16*x)^(13/8). - Ilya Gutkovskiy, Jun 07 2016
EXAMPLE
a(0) = (1 + 2 + 3 + 4) = 10;
a(1) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) = 260;
a(2) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12) = 10920;
a(3) = (1 + 2 + 3 + 4)*(5 + 6 + 7 + 8) *(9 + 10 + 11 + 12)*(13 + 14 + 15 + 16) = 633360, etc.
MATHEMATICA
FullSimplify[Table[(2^(4 n + 13/4) Gamma[1/8] Gamma[n + 13/8])/(Sqrt[Pi] Gamma[1/4]), {n, 0, 14}]]
Table[Product[16 k + 10, {k, 0, n - 1}], {n, 20}] (* Vincenzo Librandi, Feb 12 2016 *)
PROG
(Magma) [&*[(16*k+10): k in [0..n-1]]: n in [1..20]]; // Vincenzo Librandi, Feb 12 2016
(PARI) x='x+O('x^50); Vec(serlaplace(10/(1 - 16*x)^(13/8))) \\ G. C. Greubel, Apr 09 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ilya Gutkovskiy, Feb 12 2016
STATUS
approved