A004983 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k - 3).

1, -6, -6, -20, -90, -468, -2652, -15912, -99450, -640900, -4229940, -28455960, -194449060, -1346185800, -9423300600, -66591324240, -474463185210, -3404971093860, -24591457900100, -178611641590200, -1303864983608460, -9561676546462040, -70408709114856840

Offset

0,2

Links

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

Formula

G.f.: (1 - 8*x)^(3/4).

a(n) ~ -(3/4)*Gamma(1/4)^-1*n^(-7/4)*2^(3*n)*(1 + (21/32)*n^-1 + ...).

a(n) = (-8)^n/(n*Beta(n, 7/4-n)) if n > 0; a(0)=1. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 16 2004

a(n) = 8^n*Sum_{k=0..n} ((-1)^k*binomial(k-1/4,k)*binomial(n+3/4,n-k) *binomial(n+k-1,n)). - Vladimir Kruchinin, Apr 18 2016

a(n) = (-8)^n*Gamma(7/4)/(Gamma(7/4-n)*Gamma(n+1)). - Ilya Gutkovskiy, Apr 18 2016

a(n) = 8^n * Pochhammer(-3/4, n). - G. C. Greubel, Aug 22 2019

D-finite with recurrence: n*a(n) +2*(-4*n+7)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

Maple

seq(coeff(convert(series((1-8*x)^(3/4), x, 40), polynom), x, i), i=0..25); # C. Ronaldo
1, seq(2^(3*n)*(-1)^n/(n*Beta(n, 7/4-n)), n=1..25); # C. Ronaldo

Mathematica

Table[2^n/n!*Product[(4*k-3), {k, 0, n-1}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 18 2016 *)

Prog

(Maxima)
a(n):=8^n*sum((-1)^k*binomial(k-1/4, k)*binomial(n+3/4, n-k)*binomial(n+k-1, n), k, 0, n); /*  Vladimir Kruchinin, Apr 18 2016 */
(PARI) a(n) = 2^n*prod(k=0, n-1, 4*k-3)/n!; \\ Michel Marcus, Apr 18 2016
(Magma) [1] cat [2^n*&*[4*k-3: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019
(Sage) [8^n*rising_factorial(-3/4, 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-3)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

Crossrefs

Sequence in context: A045896 A309122 A115046 * A298936 A034695 A339338

Adjacent sequences: A004980 A004981 A004982 * A004984 A004985 A004986

Keyword

sign,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved