Had a strange thing happen to me today. I filed IFR HPN - HFD (White
Plains NY to Hartford CT). The route part of my clearance came back
"radar vectors Carmel[VOR], V1, Hartford[VOR], Direct".

Usually out of HPN I would expect to get the Westchester-1 departure,
but it wasn't in my clearance, and I when I read back the clearance
without it, I got "readback correct". Seemed a little strange, but I
went with the flow (I should have asked for clarification, I'm not
sure why I didn't). The turn in the SID took us away from our first
fix, so I guess I figured they were just doing us a favor or something
with a shorter routing.

We took off and I started climbing straight out. The HPN-1 has an
almost 180-degree turn almost immediately. The guy I was with asked
me why I wasn't making the turn and I said we weren't on the SID. I
asked tower and they said I should be on the SID.

So, what went wrong? Did I goof? Am I supposed to fly the SID (not a
DP) even though it's not in the clearance? Is it more likely that it
was in the clearance but I just didn't hear it and the controller
didn't notice that it wasn't in the readback?

Since you were to be given radar vectors, and tower normally gives that
first vector (or fly runway heading) you should have asked for the vector
when cleared for takeoff. However, doing so was to help them (CD/Tower) fix
their goof as one of them dropped the ball.

Not your goof; but then the PIC is always the one that runs into the
mountain - not "Them"...

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