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A143001 a(0) = 0, a(1) = 1, a(n+1) = 5*(2*n+1)*a(n) + n^4*a(n-1).

%I A143001 #9 Jan 17 2025 03:36:53
%S A143001 0,1,15,391,14900,770596,51695280,4358885616,451036788480,
%T A143001 56192122503936,8297504007091200,1433159145783936000,
%U A143001 286297057932974899200,65505120288597559296000,17020121510584366601011200,4984362320041496995061760000,1634219811079765593410764800000
%N A143001 a(0) = 0, a(1) = 1, a(n+1) = 5*(2*n+1)*a(n) + n^4*a(n-1).
%C A143001 This is the case m = 2 of the general recurrence a(0) = 0, a(1) = 1, a(n+1) = (2*m+1)*(2*n+1 )*a(n) + n^4*a(n-1) (we suppress the dependence of a(n) on m), which arises when accelerating the convergence of the series Sum_{k >= 1} (-1)^(k+1)/k^2 for the constant (1/2)*zeta(2). For other cases see A142999 (m=0), A143000(m=1) and A143002 (m=3).
%D A143001 Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag, 1989.
%F A143001 a(n) = n!^2*p(n)*Sum_{k = 1..n} (-1)^(k+1)/(k^2*p(k-1)*p(k)), where p(n) = (n^4+2*n^3+7*n^2+6*n+4)/4.
%F A143001 Recurrence: a(0) = 0, a(1) = 1, a(n+1) = 5*(2*n+1)*a(n) + n^4*a(n-1).
%F A143001 The sequence b(n):= n!^2*p(n) satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 5. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(5+1^4/(15+2^4/(25+3^4/(35+...+(n-1)^4/(5*(2*n-1)))))), for n >=2.
%F A143001 Limit_{n -> oo} a(n)/b(n) = Sum_{k >= 1} (-1)^(k+1)/(k^2*p(k-1)p(k)) = 1/(5+ 1^4/(15+ 2^4/(25+ 3^4/(35+ ... + n^4/(5*(2*n+1)+...))))) = 1/2*(zeta(2)-(1+1/4)). The final equality follows from a result of Ramanujan; see [Berndt, Chapter 12, Corollary to Entry 30].
%p A143001 p := n -> (n^4+2*n^3+7*n^2+6*n+4)/4: a := n -> n!^2*p(n)*sum ((-1)^(k+1)/(k^2*p(k-1)*p(k)), k = 1..n): seq(a(n), n = 0..20);
%t A143001 RecurrenceTable[{a[n+1] == 5*(2*n+1)*a[n] + n^4*a[n-1], a[0] == 0, a[1] == 1}, a, {n, 0, 16}] (* _Amiram Eldar_, Jan 17 2025 *)
%Y A143001 Cf. A142999, A143000, A143002.
%K A143001 easy,nonn
%O A143001 0,3
%A A143001 _Peter Bala_, Jul 18 2008
%E A143001 More terms from _Amiram Eldar_, Jan 17 2025