Album purchased in iTunes but not in library

Check for a download problem by attempting to resume download of purchased items as noted in iTunes: How to resume interrupted iTunes Store downloads

1. From the Home screen, tap the iTunes icon.

2. For iPhone/iPod touch, tap More > Downloads. For iPad, tap Downloads.

3. Enter your account name and password.

4. Click the "OK" button.

5. Click the Resume, Resume All, or button to start the download.

Note that you may want to check and see if automatic downloads are enabled on your iPad as noted here iTunes Store: How to enable Automatic Downloads this may be causing the disconnect of purchasing and not seeing any immediate downloads happening.

1. Tap Settings > Store on your device.
2. Tap Sign In if you're not already signed in to the iTunes Store.
3. Enter your Apple ID and password.
4. Tap On/Off switch to enable or disable Automatic Downloads for each content type.

If that does not work go into the iTunes Store and check your purchase history as noted in iTunes Store & Mac App Store: Seeing your purchase history and order numbers Clicking the Buy button multiple times, may be resulting in multiple purchases, so be careful and check your account purchase history.

1. Open iTunes.

2. Click iTunes Store on the left side of the iTunes window.

3. If you're not signed in to the store, click Sign In and enter your Apple ID and password.

4. When the iTunes Store opens, click the button with your account name on it, enter your password, and then click View Account.

5. In the Account Information window, under Purchase History, click See All.