A004985 a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k + 5).

1, 10, 90, 780, 6630, 55692, 464100, 3845400, 31724550, 260846300, 2138939660, 17500415400, 142920059100, 1165348174200, 9489263704200, 77179344794160, 627082176452550, 5090431785320700, 41289057814267900, 334658679126171400, 2710735300921988340, 21944047674130381800

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

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

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

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

E.g.f.: hypergeom([5/4], [1], 8*x). - Karol A. Penson, Dec 20 2015

D-finite with recurrence: n*a(n) + 2*(-4*n-1)*a(n-1) = 0. - R. J. Mathar, Jan 16 2020

Sum_{n>=0} 1/a(n) = (sqrt(2)/7^(3/4)) * (2*arccosec(2/7^(1/4)) + 2*log(7 + sqrt(7) + sqrt(2)*7^(3/4)) - log(7) - 3*log(2)). - Amiram Eldar, Dec 21 2025

Maple

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

Mathematica

Table[2^n/n! Product[4k+5, {k, 0, n-1}], {n, 0, 25}] (* Harvey P. Dale, Apr 15 2019 *)
Table[8^n*Pochhammer[5/4, n]/n!, {n, 0, 25}] (* G. C. Greubel, Aug 22 2019 *)

Prog

(PARI) a(n)=2^n/n!*prod(k=0, n-1, 4*k+5)
for(n=0, 21, print(a(n)))
(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
(SageMath) [8^n*rising_factorial(5/4, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019
(GAP) List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+5)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Cf. A004981, A004982, A004983, A004984, A004986.

Sequence in context: A162756 A331323 A199940 * A276020 A164552 A057086

Adjacent sequences: A004982 A004983 A004984 * A004986 A004987 A004988

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

More terms from Rick L. Shepherd, Mar 03 2002

Terms a(20) onward added by G. C. Greubel, Aug 22 2019

Status

approved