A047055 Quintuple factorial numbers: a(n) = Product_{k=0..n-1} (5*k + 2).
1, 2, 14, 168, 2856, 62832, 1696464, 54286848, 2008613376, 84361761792, 3965002804224, 206180145819648, 11752268311719936, 728640635326636032, 48818922566884614144, 3514962424815692218368, 270652106710808300814336, 22193472750286280666775552
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..350
Crossrefs
Programs
-
GAP
List([0..20], n-> Product([0..n-1], k-> (5*k+2) )); # G. C. Greubel, Aug 17 2019
-
Magma
[1] cat [(&*[(5*k+2): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019
-
Maple
a := n->product(5*i+2,i=0..n-1); [seq(a(j),j=0..30)];
-
Mathematica
Table[5^n*Pochhammer[2/5, n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *) Join[{1},FoldList[Times,5*Range[0,20]+2]] (* Harvey P. Dale, Apr 03 2025 *) -
PARI
vector(20, n, n--; prod(k=0,n-1, 5*k+2)) \\ G. C. Greubel, Aug 17 2019
-
Sage
[product((5*k+2) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019
Formula
E.g.f. (1-5*x)^(-2/5).
a(n) ~ sqrt(2*Pi)/Gamma(2/5)*n^(-1/10)*(5n/e)^n*(1 - (11/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = A084940(n)/A000142(n)*A000079(n) = 5^n*Pochhammer(2/5, n) = 5^n*Gamma(n+2/5)*sin(2*Pi/5)*Gamma(3/5)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
G.f.: 1/(1-2x/(1-5x/(1-7x/(1-10x/(1-12x/(1-15x/(1-17x/(1-20x/(1-22x/(1-25x/(1-.../(1-A047215(n+1)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009
a(n) = (-3)^n*Sum_{k=0..n} (5/3)^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
D-finite with recurrence: a(n) +(-5*n+3)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
G.f.: 1/G(0) where G(k) = 1 - x*(5*k+2)/( 1 - 5*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
Sum_{n>=0} 1/a(n) = 1 + (e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). - Amiram Eldar, Dec 19 2022
Comments