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A322518 Binomial transform of the Apéry numbers (A005259). 1
1, 6, 84, 1680, 39240, 999216, 26899896, 752939424, 21691531800, 638947312080, 19155738105504, 582589712312064, 17930566188602136, 557417298916695600, 17477836958370383280, 552090876791399769600, 17552554240486710112920, 561230779055361080132880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Starting with the a(3) 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.

N. J. A. Sloane, Transforms

FORMULA

a(n) ~ 2^(n - 3/4) * 3^(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) = 84.

MATHEMATICA

a[n_] := Sum[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 OEISbinomial_transform(N, seq):

    BT = [seq[0]]

    k = 1

    while k< N:

        next = 0

        j = 0

        while j <=k:

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

            j = j+1

        BT.append(next)

        k = k+1

    return BT

Apery = oeis('A005259')

OEISBinom = OEISbinomial_transform(18, Apery.first_terms(20))

CROSSREFS

Cf. A005259, A322519.

Sequence in context: A306244 A277304 A128575 * A014062 A147626 A123312

Adjacent sequences:  A322515 A322516 A322517 * A322519 A322520 A322521

KEYWORD

nonn

AUTHOR

Sarah Arpin, Dec 13 2018

STATUS

approved

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Last modified December 8 01:59 EST 2019. Contains 329850 sequences. (Running on oeis4.)