Multigraphs with a given degree sequence This challenge will give you an input of a degree sequence in the form of a partition of an even number. Your goal will be to write a program that will output the number of loop-free labeled multigraphs that have this degree sequence.

Example

For example, given the input [3,3,1,1], you should output the number of labeled multigraphs on four vertices \$\{v_1, v_2, v_3, v_4\}\$ such that \$\deg(v_1) = \deg(v_2) = 3\$ and \$\deg(v_3) = \deg(v_4) = 1\$. In this example, there are three such multigraphs:



See Gus Wiseman's links on OEIS sequence A209816 for more examples.

Target data

input           | output
----------------+--------
[6,2]           | 0
[5,3,2]         | 1
[3,3,1,1]       | 3
[4,3,1,1,1]     | 6
[4,2,1,1,1,1]   | 10
[1,1,1,1,1,1]   | 15
[3,2,2,2,1]     | 15
[2,2,2,2,2]     | 22
[7,6,5,4,3,2,1] | 7757


Challenge

This is code-golf so shortest code in bytes wins.
Haskell, 57 bytes



f(h:t)=sum[f l|l&lt;-mapM(\n-&gt;[0..n])t,sum t-sum l==h]
f _=1


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