A340571 Number of partitions of n into 4 parts with at least one even part.
0, 0, 0, 0, 0, 1, 1, 3, 3, 6, 6, 11, 10, 18, 17, 27, 25, 39, 36, 54, 49, 72, 66, 94, 85, 120, 109, 150, 135, 185, 167, 225, 202, 270, 243, 321, 287, 378, 339, 441, 394, 511, 457, 588, 524, 672, 600, 764, 680, 864, 770, 972, 864, 1089, 969, 1215, 1079, 1350, 1200, 1495, 1326
Offset: 0
Keywords
Examples
a(5) = 1; [2,1,1,1]; a(7) = 3; [4,1,1,1], [3,2,1,1], [2,2,2,1]; a(9) = 6; [6,1,1,1], [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1], [3,2,2,2].
Programs
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Mathematica
Table[Sum[Sum[Sum[1 - Mod[k, 2] Mod[j, 2] Mod[i, 2] Mod[n - i - k - j, 2], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Formula
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (1 - (k mod 2) * (j mod 2) * (i mod 2) * ((n-i-j-k) mod 2)).
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign( ((k+1) mod 2) + ((j+1) mod 2) + ((i+1) mod 2) + ((n-i-j-k+1) mod 2) ).