Is there a way to find the sum of the Kth series ( Given below)

K=0   S={1,2,3,4,5,6,....}
K=1   S={1,2,4,7,11,16..}  common diff = 1,2,3,4 5 ...
K=2   S={1,2,4,8,15,26...} common diff = 1,2,4,7 11... (series with
K=1)
K=3   S={1,2,4,8,16,31...} common diff = 1,2,4,8 15... (series with
K=2)

Note that the common difference of Kth series is the (K-1) series

Any ideas ??

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