login
a(n) = Product_{i=0..n-1} (8*i+2).
13

%I #36 Dec 20 2022 03:49:51

%S 1,2,20,360,9360,318240,13366080,668304000,38761632000,2558267712000,

%T 189311810688000,15523568476416000,1397121162877440000,

%U 136917873961989120000,14513294639970846720000,1654515588956676526080000,201850901852714536181760000,26240617240852889703628800000

%N a(n) = Product_{i=0..n-1} (8*i+2).

%H G. C. Greubel, <a href="/A084948/b084948.txt">Table of n, a(n) for n = 0..330</a>

%F a(n) = A084943(n)/A000142(n)*A000079(n) = 8^n*Pochhammer(1/4, n) = 1/2*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)*8^n/Pi.

%F a(n) = (-6)^n*Sum_{k=0..n} (4/3)^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 G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+2)/(2*x*(8*k+2) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 30 2013

%F From _Ilya Gutkovskiy_, Mar 23 2017: (Start)

%F E.g.f.: 1/(1 - 8*x)^(1/4).

%F a(n) ~ sqrt(2*Pi)*8^n*n^n/(exp(n)*n^(1/4)*Gamma(1/4)). (End)

%F Sum_{n>=0} 1/a(n) = 1 + (e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). - _Amiram Eldar_, Dec 20 2022

%p a := n->product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

%t Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* _G. C. Greubel_, Aug 18 2019 *)

%o (PARI) vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ _G. C. Greubel_, Aug 18 2019

%o (Magma) [1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // _G. C. Greubel_, Aug 18 2019

%o (Sage) [product(8*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-> 8*k+2) ); # _G. C. Greubel_, Aug 18 2019

%Y Cf. A000079, A000142, A000165, A008544, A001813, A047055, A047657, A048994, A084943, A084947, A084949.

%K easy,nonn

%O 0,2

%A Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003