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%I #9 Aug 09 2013 14:24:26
%S 5,3,5,6,3,6,1,9,4,7,8,0,7,8,7,2,8,4,5,5,7,8,5,0,7,4,8,6,6,4,7,1,8,6,
%T 0,7,0,0,4,3,5,4,9,4,9,6,9,1,0,7,9,6,2,2,7,7,5,5,2,0,9,8,4,9,1,8,8,7,
%U 9,3,3,8,3,3,6,0,7,1,3,2,4,9,7,9,7,8
%N Decimal expansion of sum of reciprocals, row 4 of the natural number array, A185787.
%C 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 Let c(n) be the sum of reciprocals of the numbers in column n 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.
%e 1/10 + 1/14 + 1/19 + ... = (1/4340)*(-2573 + 560r*tanh(r/2), where r=(pi/2)sqrt(31)
%e 1/10 + 1/14 + 1/19 + ... = 0.5356361947807872845578507486647...
%t $MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[4, k], {k, 1, Infinity}], 130]; RealDigits[u, 10]
%Y Cf. A185787, A000027, A228044, A226985.
%K nonn,cons,easy
%O 0,1
%A _Clark Kimberling_, Aug 06 2013