A004985 - OEIS

%I #32 Sep 08 2022 08:44:33

%S 1,10,90,780,6630,55692,464100,3845400,31724550,260846300,2138939660,

%T 17500415400,142920059100,1165348174200,9489263704200,77179344794160,

%U 627082176452550,5090431785320700,41289057814267900,334658679126171400,2710735300921988340,21944047674130381800

%N a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k + 5).

%H G. C. Greubel, <a href="/A004985/b004985.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (1 - 8*x)^(-5/4).

%F a(n) ~ 4*Gamma(1/4)^-1*n^(1/4)*2^(3*n)*{1 + 5/32*n^-1 - ...}

%F a(n) = 8^n*binomial(1/4 + n, 1/4).

%F E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([5/4], [1], 8*x). - _Karol A. Penson_, Dec 20 2015

%F D-finite with recurrence: n*a(n) +2*(-4*n-1)*a(n-1)=0. - _R. J. Mathar_, Jan 16 2020

%p a:= n-> (2^n/n!)*mul(4*k+5, k=0..n-1); seq(a(n), n=0..25); # _G. C. Greubel_, Aug 22 2019

%t Table[2^n/n! Product[4k+5,{k,0,n-1}],{n,0,25}] (* _Harvey P. Dale_, Apr 15 2019 *)

%t Table[8^n*Pochhammer[5/4, n]/n!, {n,0,25}] (* _G. C. Greubel_, Aug 22 2019 *)

%o (PARI) a(n)=2^n/n!*prod(k=0,n-1,4*k+5)

%o for(n=0,21,print(a(n)))

%o (Magma) [1] cat [2^n*&*[4*k+5: k in [0..n-1]]/Factorial(n): n in [1..25]]; // _G. C. Greubel_, Aug 22 2019

%o (Sage) [8^n*rising_factorial(5/4, n)/factorial(n) for n in (0..25)] # _G. C. Greubel_, Aug 22 2019

%o (GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+5)/Factorial(n) ); # _G. C. Greubel_, Aug 22 2019

%K nonn,easy

%O 0,2

%A Joe Keane (jgk(AT)jgk.org)

%E More terms from _Rick L. Shepherd_, Mar 03 2002

%E Terms a(20) onward added by _G. C. Greubel_, Aug 22 2019