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FORMULA
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a(n) = Sum_{i=0..n-1} (-1)^i*(2+i)!*Stirling2(n,2+i)*Catalan(2,i)/2!, where Stirling2(n,k) = A008277(n,k); Catalan(k,i) = binomial(2*i+k,i)*k/(2*i+k) = coefficient of x^i in C(x)^k with C(x) = (1-sqrt(1-4x))/(2x).
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PROG
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(PARI) {a(n)=n!* sum(i=0, n-1, (-1)^i*polcoeff(((exp(x+x*O(x^n))-1)^(2+i)), n)*binomial(2*i+2, i)/(2*i+2))}
for(n=2, 20, print1(a(n), ", "))
(PARI) /* Define Stirling2: */
{Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
/* Define Catalan(m, n) = [x^n] C(x)^m: */
{Catalan(m, n)=binomial(2*n+m, n)*m/(2*n+m)}
/* Define this sequence: */
{a(n)=sum(i=0, n-1, (-1)^i*(2+i)!*Stirling2(n, 2+i)*Catalan(2, i)/2!)}
for(n=2, 20, print1(a(n), ", "))
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