The series is: 2^1*3^0, 2^3*3^1, 2^4*3^2, 2^7*3^3.
The powers of 3 are progressive: 0, 1, 2, 3, etc., but the powers of 2 are: 1, 3, 4, 7. It would seem that 3^n is part of the explicit formula, but we also need a formula for the powers of 2.
It appears that the the sum of the first 2 powers of 2 equals the third (1+3=4) and the sum of the next pair equals the next power (3+4=7). The next power using this pattern would be 11. However, there are insufficient terms to determine whether this pattern is the actual one. The series 1, 3, 4, 7 is related to the Fibonacci series, and there is no explicit formula for the nth term without invoking an irrational expression, which is unlikely to be a solution for this question. (A more plausible formula would be: 2^(2n+1)3^n or 2*12^n, which gives 2, 24, 288, 3456, 41472, etc.)
There is a formula relating 1 3 4 and 7 as the first few terms of a series: n^3/2-2n^2+7n/2+1. When n=0, we have 1; n=1, 3; n=2, 4; n=3, 7; n=4, 15; n=5, 31; n=6, 58; n=7, 99; n=8, 157. So the explicit formula would be for the nth term, starting at n=0: 2^(n^3/2-2n^2+7n/2+1)*3^n. This is an unlikely formula which suggests that the series may have been wrongly presented.