login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A322519 Inverse binomial transform of the Apéry numbers (A005259). 1
1, 4, 64, 1240, 27640, 667744, 17013976, 450174736, 12250723480, 340711148320, 9641274232384, 276704848753216, 8035189363318936, 235655550312118720, 6970100090159566480, 207674717284507191520, 6227433643414033714840, 187795334412416019255520 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Starting with the a(2) term, each term is divisible by 8. (Empirical observation.)

LINKS

Table of n, a(n) for n=0..17.

Jackson Earles, Justin Ford, Poramate Nakkirt, Marlo Terr, Dr. Ilia Mishev, Sarah Arpin, Binomial Transforms of Sequences, Fall 2018.

FORMULA

a(n) = Sum_{i=0..n} C(n,i) * (-1)^i * A005259(n-i).

a(n) ~ 2^((5*n + 3)/2) * (1 + sqrt(2))^(2*n - 1) / (Pi*n)^(3/2). - Vaclav Kotesovec, Dec 17 2018

EXAMPLE

a(2) = binomial(2,0)*A(0) - binomial(2,1)*A(1) + binomial(2,2)*A(2), where A(k) denotes the k-th Apéry number. Using this definition:

a(2) = binomial(2,0)*(binomial(0,0)*binomial(0,0))^2 - binomial(2,1)*((binomial(1,0)*binomial(1,0))^2 + (binomial(1,1)*binomial(2,1))^2) + binomial(2,2)*((binomial(2,0)*binomial(2,0))^2 + (binomial(2,1)*binomial(3,1))^2 + (binomial(2,2)*binomial(4,2))^2) = 64.

MAPLE

a:=n->add(binomial(n, i)*(-1)^i*add((binomial(n-i, k)*binomial(n-i+k, k))^2, k=0..n-i), i=0..n): seq(a(n), n=0..20); # Muniru A Asiru, Dec 22 2018

MATHEMATICA

a[n_] := Sum[(-1)^(n-k) * Binomial[n, k] * Sum[(Binomial[k, j] * Binomial[k+j, j])^2, {j, 0, k}], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 13 2018 *)

PROG

(Sage)

def OEISInverse(N, seq):

    BT = [seq[0]]

    k = 1

    while k< N:

        next = 0

        j = 0

        while j <=k:

            next = next + (((-1)^(j+k))*(binomial(k, j))*seq[j])

            j = j+1

        BT.append(next)

        k = k+1

    return BT

Apery = oeis('A005259')

OEISInverse(18, Apery)

(Sage) [sum((-1)^(n-k)*binomial(n, k)*sum((binomial(k, j)* binomial(k+j, j))^2 for j in (0..k)) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Dec 13 2018

(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*sum(j=0, k, (binomial(k, j)*binomial(k+j, j))^2))};

for(n=0, 20, print1(a(n), ", ")) \\ G. C. Greubel, Dec 13 2018

(MAGMA) [(&+[(-1)^(n-k)*Binomial(n, k)*(&+[(Binomial(k, j)*Binomial(k+j, j))^2: j in [0..k]]): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 13 2018

CROSSREFS

Cf. A005259, A322518.

Sequence in context: A152923 A085807 A014729 * A259272 A085532 A146341

Adjacent sequences:  A322516 A322517 A322518 * A322520 A322521 A322524

KEYWORD

nonn

AUTHOR

Sarah Arpin, Dec 13 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 20 02:06 EDT 2019. Contains 322291 sequences. (Running on oeis4.)