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A004982 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3). 5
1, 6, 42, 308, 2310, 17556, 134596, 1038312, 8046918, 62587140, 488179692, 3816677592, 29897307804, 234578876616, 1843119744840, 14499208659408, 114181268192838, 900017055167076, 7100134546318044, 56053693786721400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

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.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [1], 8*x).

Representation as n-th moment of a positive function on (0, 8): a(n)=int(x^n*(2^(1/4)/(8*Pi*x^(1/4)*(1-x/8)^(3/4)), x=0..8), n=0, 1, ... . 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)

MAPLE

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

seq(A004982(n), n=0..19); # Peter Luschny, Oct 23 2018

MATHEMATICA

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, 20}] (* Vaclav Kotesovec, Sep 15 2013 *)

FullSimplify[Table[8^n*Gamma[n+3/4]/(n!*Gamma[3/4]), {n, 0, 20}]] (* Vaclav Kotesovec, Sep 15 2013 *)

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

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

PROG

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

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

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

CROSSREFS

Main diagonal of A067001.

Sequence in context: A111602 A299916 A091164 * A093388 A162968 A247638

Adjacent sequences:  A004979 A004980 A004981 * A004983 A004984 A004985

KEYWORD

nonn,easy

AUTHOR

Joe Keane (jgk(AT)jgk.org)

EXTENSIONS

More terms from Rick L. Shepherd, Mar 03 2002

STATUS

approved

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Last modified January 20 04:43 EST 2019. Contains 319323 sequences. (Running on oeis4.)