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

%I A322519 #30 Sep 08 2022 08:46:23
%S A322519 1,4,64,1240,27640,667744,17013976,450174736,12250723480,340711148320,
%T A322519 9641274232384,276704848753216,8035189363318936,235655550312118720,
%U A322519 6970100090159566480,207674717284507191520,6227433643414033714840,187795334412416019255520
%N A322519 Inverse binomial transform of the Apéry numbers (A005259).
%C A322519 Starting with the a(2) term, each term is divisible by 8. (Empirical observation.)
%H A322519 Jackson Earles, Justin Ford, Poramate Nakkirt, Marlo Terr, Dr. Ilia Mishev, Sarah Arpin, <a href="https://www.colorado.edu/math/binomial-transforms-sequences-fall-2018">Binomial Transforms of Sequences</a>, Fall 2018.
%F A322519 a(n) = Sum_{i=0..n} C(n,i) * (-1)^i * A005259(n-i).
%F A322519 a(n) ~ 2^((5*n + 3)/2) * (1 + sqrt(2))^(2*n - 1) / (Pi*n)^(3/2). - _Vaclav Kotesovec_, Dec 17 2018
%e A322519 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:
%e A322519 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.
%p A322519 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
%t A322519 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 *)
%o A322519 (Sage)
%o A322519 def OEISInverse(N, seq):
%o A322519     BT = [seq[0]]
%o A322519     k = 1
%o A322519     while k< N:
%o A322519         next = 0
%o A322519         j = 0
%o A322519         while j <=k:
%o A322519             next = next + (((-1)^(j+k))*(binomial(k,j))*seq[j])
%o A322519             j = j+1
%o A322519         BT.append(next)
%o A322519         k = k+1
%o A322519     return BT
%o A322519 Apery = oeis('A005259')
%o A322519 OEISInverse(18,Apery)
%o A322519 (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
%o A322519 (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))};
%o A322519 for(n=0, 20, print1(a(n), ", ")) \\ _G. C. Greubel_, Dec 13 2018
%o A322519 (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
%Y A322519 Cf. A005259, A322518.
%K A322519 nonn
%O A322519 0,2
%A A322519 _Sarah Arpin_, Dec 13 2018