2 added 193 characters in body
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Mathematica, 5959 57 bytes

Thanks to Martin Ender for finding a two-byte savings!

Table[Sum[Binomial[iBinomial[i,(j+i)/2],2]~Sum~{i,Abs@j,b,2}],~Table~{j,-b,b=#-1}]&&

Pure function taking a positive integer input and returning a list of integers. Literally produces all the relevant entries of Pascal's triangle and sums them appropriately.

Previous submission (which is a bit easier to read):

Table[Sum[Binomial[i,(j+i)/2],{i,Abs@j,b,2}],{j,-b,b=#-1}]&

Mathematica, 59 bytes

Table[Sum[Binomial[i,(j+i)/2],{i,Abs@j,b,2}],{j,-b,b=#-1}]&

Pure function taking a positive integer input and returning a list of integers. Literally produces all the relevant entries of Pascal's triangle and sums them appropriately.

Mathematica, 59 57 bytes

Thanks to Martin Ender for finding a two-byte savings!

Binomial[i,(j+i)/2]~Sum~{i,Abs@j,b,2}~Table~{j,-b,b=#-1}&

Pure function taking a positive integer input and returning a list of integers. Literally produces all the relevant entries of Pascal's triangle and sums them appropriately.

Previous submission (which is a bit easier to read):

Table[Sum[Binomial[i,(j+i)/2],{i,Abs@j,b,2}],{j,-b,b=#-1}]&
1
source | link

Mathematica, 59 bytes

Table[Sum[Binomial[i,(j+i)/2],{i,Abs@j,b,2}],{j,-b,b=#-1}]&

Pure function taking a positive integer input and returning a list of integers. Literally produces all the relevant entries of Pascal's triangle and sums them appropriately.