Songbird intercept and differentials interpretation in multivariable model

I have a question regarding the intercepts and differentials I got from Songbird using the following formula "C(Sampletype, Diff, levels=['3', '2', '1'])+Penstyle"

I have three sample types (1, 2, 3) and two pen styles (old and new). The way the formula is written, sample type 1 and new pens are the reference. In this way, there are differentials created for sample types 2 and 3 and old pens.

I understand how to interpret the intercept and differentials in cases that only look at one categorical variable, however when there are multiple in the model I am having trouble understanding.

Is the intercept value = (mean sample type 1 + mean new pen ) + (mean sample type 2 - mean sample type 1) + (mean sample type 3 - mean sample type 1) + (mean old pen - mean new)??

Also, in terms of the differentials - does the differential for sample type 2 indicate it's log-fold change compared to sample type 1 by itself or to the reference which is sample type 1 in new pens.

Please let me know if I have been unclear on anything. I want to make sure that I am interpreting this correctly, thanks in advance for all your help!!

Regarding the differentials, its the (biased) log-fold change between sample 2 and sample 3 (that's because of the Diff argument, it is a backwards encoding that takes the difference between previous levels). We are using patsy, so it is possible verify this yourself with the patsy tutorials (see here).

Regarding the intercept, it should just be the mean of sample 3 (which is the reference implied by the backwards difference encoding).