0,2

Conjecture: a(p*n) = a(n) (mod p^2) for prime p == 1 (mod 4) and all positive integers n. Cf. A004981. - Peter Bala, Dec 22 2019

Alois P. Heinz, Table of n, a(n) for n = 0..400

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

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

a(n) = 8^n*Gamma(n+3/4)/(n!*Gamma(3/4)). - Vaclav Kotesovec, Sep 15 2013

From Karol A. Penson, Dec 19 2015: (Start)

a(n) = (-8)^n*binomial(-3/4,n).

E.g.f.: hypergeom([3/4], [1], 8*x).

Representation as n-th moment of a positive function on (0, 8): a(n) = Integral_{x=0..8} ( x^n*2^(1/4)/(8*Pi*x^(1/4)*(1-x/8)^(3/4)) ) dx, n >= 0. This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)

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) = 8/7 + (sqrt(2)/7^(5/4))*(2*arccosec(2/7^(1/4)) - arccosh(sqrt(11+4*sqrt(7))/2)). - Amiram Eldar, Dec 21 2025

A004982 := n -> (-8)^n*binomial(-3/4, n):

seq(A004982(n), n=0..25); # Peter Luschny, Oct 23 2018

Table[2^n/n! Product[4k+3, {k, 0, n-1}], {n, 0, 30}] (* Harvey P. Dale, Oct 03 2011 *)

Table[Sum[2^k*Binomial[2*n-2*k, n-k]*Binomial[n+k, n], {k, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Sep 15 2013 *)

FullSimplify[Table[8^n*Gamma[n+3/4]/(n!*Gamma[3/4]), {n, 0, 25}]] (* Vaclav Kotesovec, Sep 15 2013 *)

max = 30; s = Hypergeometric1F1[3/4, 1, 8x] + O[x]^(max+1);

CoefficientList[s, x]*(Range[0, max]!) (* Jean-François Alcover, Dec 19 2015, after Karol A. Penson *)

(PARI) a(n)=2^n/n!*prod(k=0, n-1, 4*k+3)

for(n=0, 25, print(a(n)))

(PARI) my(x='x+O('x^66)); Vec((1-8*x)^(-3/4)) \\ Joerg Arndt, Apr 20 2013

(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

(SageMath) [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

Main diagonal of A067001.

Cf. A004981, A004983, A004984, A004985, A004986.

Sequence in context: A111602 A299916 A091164 * A093388 A162968 A247638

Adjacent sequences: A004979 A004980 A004981 * A004983 A004984 A004985

nonn,easy

Joe Keane (jgk(AT)jgk.org)

More terms from Rick L. Shepherd, Mar 03 2002

approved