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
Keywords
Examples
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.
Links
- Jackson Earles, Justin Ford, Poramate Nakkirt, Marlo Terr, Dr. Ilia Mishev, Sarah Arpin, Binomial Transforms of Sequences, Fall 2018.
Programs
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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
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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
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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 *) -
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 -
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
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
Comments