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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
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
Sequence in context: A152923 A085807 A014729 * A259272 A361963 A085532
KEYWORD
nonn
AUTHOR
Sarah Arpin, Dec 13 2018
STATUS
approved