OFFSET
0,2
COMMENTS
Starting with the a(2) term, each term is divisible by 8. (Empirical observation.)
LINKS
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
KEYWORD
nonn
AUTHOR
Sarah Arpin, Dec 13 2018
STATUS
approved