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2 added 361 characters in body
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05AB1E, 99 8 bytes

Code:Thanks to Emigna for saving a byte!

Code:

LO;¹LæOQOLæO·sLO¢

Uses the 05AB1E encoding. Try it online!Try it online!

Explanation

L           # Create the list [1, 2, .., input]
 æ          # Compute the powerset of this list
  O         # Sum each list
   ·        # Double each element
    sLO     # Compute the sum of [1, 2, .., input]
       ¢    # Count the number of occurrences

05AB1E, 9 bytes

Code:

LO;¹LæOQO

Uses the 05AB1E encoding. Try it online!

05AB1E, 9 8 bytes

Thanks to Emigna for saving a byte!

Code:

LæO·sLO¢

Uses the 05AB1E encoding. Try it online!

Explanation

L           # Create the list [1, 2, .., input]
 æ          # Compute the powerset of this list
  O         # Sum each list
   ·        # Double each element
    sLO     # Compute the sum of [1, 2, .., input]
       ¢    # Count the number of occurrences
1
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05AB1E, 9 bytes

Code:

LO;¹LæOQO

Uses the 05AB1E encoding. Try it online!