About a New Kind of Ramanujan-Type Series

where d, k, a, b, c are integers, B(n) = n!−5 C(n) or B(n) = (−1)nn!−5C(n), and C(n) is the product of 5 rising factorials of fractions smaller than unity satisfying the following condition: For every denominator in the fraction of a rising factorial, we must have rising factorials with all possible nonreducible fractions corresponding to that denominator. Taking this into account, we have the following cases for C(n): ( 1 2 )