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A228044 Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.

%I A228044 #21 Feb 23 2025 22:57:05
%S A228044 1,1,2,2,2,9,4,6,0,6,6,0,3,5,0,4,3,4,3,5,4,3,4,3,2,1,8,5,9,7,9,2,5,5,
%T A228044 9,9,2,0,2,4,3,5,0,0,8,4,2,6,9,4,6,5,5,6,7,8,8,6,4,8,1,7,3,4,3,0,8,9,
%U A228044 9,0,3,8,1,2,1,3,5,0,3,9,6,5,8,1,0,2
%N A228044 Decimal expansion of sum of reciprocals, row 2 of the natural number array, A185787.
%C A228044 Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)*(n+k-1)/2, and let r = (2*Pi/sqrt(7))*tanh(Pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047.
%C A228044 Let c(k) be the sum of reciprocals of the numbers in column k of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*Pi^2; d(3) = 1/2, and d(1) is given by A228048.
%C A228044 It appears that Mathematica gives closed-form exact expressions for s(n), c(n) for 1<=n<=20 and further. The same holds for diagonal sums dr(n,n+k), k>=0; and for diagonal sums and dc(n+k,n), k>=0. In any case, general terms for all four sequences can be represented using the digamma function. The representations imply that c(n) is rational if and only if n is a term of A000124, and that dr(n) is rational if and only if n has the form k^2 + 2 for k >= 1.
%F A228044 Equals 1/3 + 1/5 + 1/8 + ...
%F A228044 Equals (1/30)*(-15 + 8*r*tanh(r)), where r = (Pi/2)*sqrt(15).
%e A228044 1.12229460660350434354343218597925...
%t A228044 $MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2;
%t A228044 u = N[Sum[1/t[2, k], {k, 1, Infinity}], 130]
%t A228044 RealDigits[u, 10]
%o A228044 (PARI) sumnumrat(2/(n*(n+1)+4),1) \\ _Charles R Greathouse IV_, Feb 08 2023
%Y A228044 Cf. A185787, A000027, A228040, A226985, A228045.
%K A228044 nonn,cons,easy
%O A228044 1,3
%A A228044 _Clark Kimberling_, Aug 06 2013