This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A061572 #18 Dec 23 2024 13:09:07
%S A061572 1,5,47,758,18974,683184,33476736,2142516144,173543847984,
%T A061572 17354385161280,2099880608143680,302382807612606720,
%U A061572 51102694487009537280,10016128119460096327680,2253628826878608852019200,576928979680925173791283200,166732475127787396148470732800
%N A061572 a(n) = (n!)^2 * Sum_{k=1..n} 1/(k^2*(k-1)!).
%H A061572 Harry J. Smith, <a href="/A061572/b061572.txt">Table of n, a(n) for n = 1..100</a>
%F A061572 Recurrence: a(1) = 1, a(2) = 5, a(n) = (n^2+n-1)*a(n-1) - (n-1)^3*a(n-2) for n >= 3. The sequence b(n) = n!^2 also satisfies this recurrence with the initial conditions b(1) = 1 and b(2) = 4. Hence we have the finite continued fraction expansion a(n)/b(n) = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)))). Lim n -> infinity a(n)/b(n) = Ei(1) - gamma = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)-...))). Cf. A061573. - _Peter Bala_, Jul 10 2008
%o A061572 (PARI) a(n) = { n!^2*sum(k=1, n, 1/(k^2*(k-1)!)) } \\ _Harry J. Smith_, Jul 24 2009
%Y A061572 Cf. A061573.
%K A061572 nonn
%O A061572 1,2
%A A061572 _N. J. A. Sloane_, May 19 2001