The agg function will do this for you. Pass the columns and function as a dict with column, output:
df.groupby(['Country', 'Item_Code']).agg({'Y1961': np.sum, 'Y1962': [np.sum, np.mean]}) # Added example for two output columns from a single input column
This will display only the group by columns, and the specified aggregate columns. In this example I included two agg functions applied to 'Y1962'.
To get exactly what you hoped to see, included the other columns in the group by, and apply sums to the Y variables in the frame:
df.groupby(['Code', 'Country', 'Item_Code', 'Item', 'Ele_Code', 'Unit']).agg({'Y1961': np.sum, 'Y1962': np.sum, 'Y1963': np.sum})