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a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(n^2 + 5n5*n + 5)/720.
a(n) = (n+1)()*(n+2)^2*(n+3)^2*(n+4)()*(n^2 + 5n + 5)/720.
Sum_{n>=0} 1/a(n) = 120*Pi^2 - 144*sqrt(5)*Pi*tan(sqrt(5)*Pi/2) - 790. - Amiram Eldar, May 31 2022
Table[(n+1)(n+2)^2(n+3)^2(n+4)(n^2+5n+5)/720, {n, 0, 30}] (* or *) LinearRecurrence[{9, -36, 84, -126, 126, -84, 36, -9, 1}, {1, 22, 190, 1015, 4018, 12936, 35784, 88110, 197835}, 30] (* Harvey P. Dale, Sep 27 2020 *)