A322518 Binomial transform of the Apéry numbers (A005259).

1, 6, 84, 1680, 39240, 999216, 26899896, 752939424, 21691531800, 638947312080, 19155738105504, 582589712312064, 17930566188602136, 557417298916695600, 17477836958370383280, 552090876791399769600, 17552554240486710112920, 561230779055361080132880

Offset

0,2

Comments

Starting with the a(3) term, each term is divisible by 8. (Empirical observation.)

The above is true and follows easily from the pair of known congruences for the Apéry numbers A(n): A(2*n) == 1 (mod 8) and A(2n+1) == 5 (mod 8). - Peter Bala, Jan 06 2020

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

The Gauss congruences hold: a(n*p^k) == a(n*p^(k-1)) (mod p^k) for all primes p and n a positive integer. - Peter Bala, Jan 06 2020

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))
(Julia)
function BinomialTransform(seq)
    N = length(seq)
    bt = Array{BigInt, 1}(undef, N)
    bt[1] = seq[1]
    for k in 1:N-1
        next = BigInt(0)
        for j in 0:k next += binomial(k, j)*seq[j+1] end
        bt[k+1] = next
    end
bt end
BinomialTransform([A005259(n) for n in 0:18]) |> println # Peter Luschny, Jan 06 2020

Crossrefs

Cf. A005259, A322519.

Sequence in context: A277304 A128575 A369531 * A014062 A147626 A123312

Adjacent sequences: A322515 A322516 A322517 * A322519 A322520 A322521

Keyword

nonn,easy

Author

Sarah Arpin, Dec 13 2018

Status

approved