A084948 a(n) = Product_{i=0..n-1} (8*i+2).

1, 2, 20, 360, 9360, 318240, 13366080, 668304000, 38761632000, 2558267712000, 189311810688000, 15523568476416000, 1397121162877440000, 136917873961989120000, 14513294639970846720000, 1654515588956676526080000, 201850901852714536181760000, 26240617240852889703628800000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..330

Crossrefs

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

Programs

GAP

List([0..20], n-> Product([0..n-1], k-> 8*k+2) ); # G. C. Greubel, Aug 18 2019

Magma

[1] cat [(&*[8*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

Maple

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

Mathematica

Table[8^n*Pochhammer[1/4, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

PARI

vector(20, n, n--; prod(k=0, n-1, 8*k+2)) \\ G. C. Greubel, Aug 18 2019

Sage

[product(8*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

Formula

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.

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

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

From Ilya Gutkovskiy, Mar 23 2017: (Start)

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

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

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