A322519 Inverse binomial transform of the Apéry numbers (A005259).

1, 4, 64, 1240, 27640, 667744, 17013976, 450174736, 12250723480, 340711148320, 9641274232384, 276704848753216, 8035189363318936, 235655550312118720, 6970100090159566480, 207674717284507191520, 6227433643414033714840, 187795334412416019255520

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 A361963 A085532

Adjacent sequences: A322516 A322517 A322518 * A322520 A322521 A322522

Keyword

nonn

Author

Sarah Arpin, Dec 13 2018

Status

approved