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a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Binomial[i, l] Sum[(-1)^j Binomial[i - l, j] Binomial[-l + 4(-l - 2j + i) - j + i - 1, 4(-l - 2j + i) - j], {j, 0, (4(i - l))/9}] (-1)^l, {l, 0, i}] a[n - i], {i, 1, n}]/n];
a /@ Range[1, 21] (* Jean-François Alcover, Sep 24 2019, after Vladimir Kruchinin *)
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a(n):=if n=0 then 1 else sum(sum(binomial(i, l)*sum((-1)^j*binomial(i-l, j)*binomial(-l+4*(-l-2*j+i)-j+i-1, 4*(-l-2*j+i)-j), j, 0, (4*(i-l))/9)*(-1)^l, l, 0, i)*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin , Apr 07 2017 */
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a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0..(4*(i-l))/9}((-1)^j*binomial(i-l,j)*binomial(-l+4*(-l-2*j+i)-j+i-1,4*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. _- _Vladimir Kruchinin_, Apr 07 2017
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a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0..(4*(i-l))/9}((-1)^j*binomial(i-l,j)*binomial(-l+4*(-l-2*j+i)-j+i-1,4*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. Vladimir Kruchinin, Apr 07 2017
(Maxima)
a(n):=if n=0 then 1 else sum(sum(binomial(i, l)*sum((-1)^j*binomial(i-l, j)*binomial(-l+4*(-l-2*j+i)-j+i-1, 4*(-l-2*j+i)-j), j, 0, (4*(i-l))/9)*(-1)^l, l, 0, i)*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin Apr 07 2017 */
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