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Number of partitions of n into 4 parts whose largest 3 parts have the same parity.
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%I #7 Jan 19 2021 21:27:50

%S 0,0,0,0,1,0,1,1,3,1,4,3,7,4,9,7,14,9,17,14,24,17,29,24,38,29,45,38,

%T 57,45,66,57,81,66,93,81,111,93,126,111,148,126,166,148,192,166,214,

%U 192,244,214,270,244,305,270,335,305,375,335,410,375,455,410,495,455,546,495

%N Number of partitions of n into 4 parts whose largest 3 parts have the same parity.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%H <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,2,-1,-2,-2,-1,2,1,1,0,-1).

%F a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} [(j mod 2) = (i mod 2) = ((n-i-j-k) mod 2)], where [ ] is the (generalized) Iverson bracket.

%F G.f.: x^4/((x^2+x+1)*(x^2+1)^2*(x+1)^3*(x-1)^4). - _Alois P. Heinz_, Jan 19 2021

%t Table[Sum[Sum[Sum[KroneckerDelta[Mod[i, 2], Mod[j, 2], Mod[n - i - j - k, 2]], {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 80}]

%K nonn,easy

%O 0,9

%A _Wesley Ivan Hurt_, Jan 19 2021