The infinite sum of the reciprocals of the squares of the positive triangular numbers is Sum_{k>=1} 1/(k*(k+1)/2)^2 = 1/1^2 + 1/3^2 + 1/6^2 + 1/10^2 + ... = 4*Pi^2/3 - 12 (see A340216).
For large values of j, the finite sum Sum_{k=1..j} 1/(k*(k+1)/2)^2 = 1/1^2 + 1/3^2 + 1/6^2 + ... + 1/(j*(j+1)/2)^2 approaches 4*Pi^2/3 - 12 - (4/3)/j^3 + 4/j^4 - (116/15)/j^5 + 12/j^6 - (340/21)/j^7 + 20/j^8 - ...; this can be written as c_0 + c_1/j + c_2/j^2 + c_3/j^3 + ... + c_m/j^m + ... where the coefficients are as follows:
c_0 = 4*Pi^2/3 - 12
c_1 = 0 c_2 = 0
c_3 = -4/3 c_4 = 4
c_5 = -116/15 c_6 = 12
c_7 = -340/21 c_8 = 20
c_9 = -356/15 c_10 = 28
c_11 = -1076/33 c_12 = 36
c_13 = -51836/1365 c_14 = 44
c_15 = -172/3 c_16 = 52
c_17 = 188/255 c_18 = 60
c_19 = -201004/399 c_20 = 68
c_21 = 686564/165 c_22 = 76
c_23 = -3423572/69 c_24 = 84
c_25 = 945336244/1365 c_26 = 92
c_27 = -34212700/3 c_28 = 100
c_29 = 94997798876/435 c_30 = 108
c_31 = -34463365906052/7161 c_32 = 116
c_33 = 30837284134268/255 c_34 = 124
c_35 = -10310751433852/3 c_36 = 132
c_37 = 105261086212083404572/959595 c_38 = 140
c_39 = -11719975655366668/3 c_40 = 148
c_41 = 1044330873985795459924/6765 c_42 = 156
c_43 = -6080390575672283355244/903 c_44 = 164
...
For even m > 2, c_m = 4*m - 12; for odd m = 2n+1, c_m appears to be a rational fraction with denominator A001897(n).
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