A004986 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 7).

1, 14, 154, 1540, 14630, 134596, 1211364, 10729224, 93880710, 813632820, 6997242252, 59794615608, 508254232668, 4300612737960, 36248021648520, 304483381847568, 2550048322973382, 21300403638954132, 177503363657951100, 1476080603050330200, 12251469005317740660, 101512171758346994040

Offset

0,2

Links

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

Formula

G.f.: (1 - 8*x)^(-7/4).

a(n) ~ 4/3*Gamma(3/4)^-1*n^(3/4)*2^(3*n)*(1 + 21/32*n^-1 - ...).

D-finite with recurrence: n*a(n) + 2*(-4*n-3)*a(n-1) = 0. - R. J. Mathar, Jan 17 2020

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

Maple

seq(2^n/n!*mul(4*k + 7, k=0..n-1), n=0..30);

Mathematica

Table[2^n/n! Product[4k+7, {k, 0, n-1}], {n, 0, 25}] (* Harvey P. Dale, Apr 26 2019 *)
Table[8^n*Pochhammer[7/4, n]/n!, {n, 0, 25}] (* G. C. Greubel, Aug 22 2019 *)

Prog

(PARI) a(n) = 2^n*prod(k=0, n-1, 4*k+7)/n!;
vector(25, n, n--; a(n)) \\ G. C. Greubel, Aug 22 2019
(Magma) [1] cat [2^n*(&*[4*k+7: k in [0..n-1]])/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
(SageMath) [8^n*rising_factorial(7/4, 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+7)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Cf. A004981, A004982, A004983, A004984, A004985.

Sequence in context: A229315 A257288 A125426 * A154248 A006865 A391779

Adjacent sequences: A004983 A004984 A004985 * A004987 A004988 A004989

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

More terms from Sascha Kurz, Mar 24 2002

Status

approved