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Number of partitions satisfying cn(0,5) <= cn(1,5) + cn(2,5) and cn(0,5) <= cn(4,5) + cn(2,5) and cn(0,5) <= cn(1,5) + cn(3,5) and cn(0,5) <= cn(4,5) + cn(3,5).
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%I #17 Oct 11 2024 12:53:08

%S 1,1,2,3,5,6,10,13,19,25,36,46,64,82,110,144,188,239,311,393,506,635,

%T 805,1004,1263,1567,1951,2408,2976,3648,4485,5471,6682,8113,9854,

%U 11907,14398,17325,20842,24984,29930,35743,42662,50758,60354,71564,84817,100267,118431,139579,164340,193181,226840,265866,311345

%N Number of partitions satisfying cn(0,5) <= cn(1,5) + cn(2,5) and cn(0,5) <= cn(4,5) + cn(2,5) and cn(0,5) <= cn(1,5) + cn(3,5) and cn(0,5) <= cn(4,5) + cn(3,5).

%C For a given partition cn(i,n) means the number of its parts equal to i modulo n.

%C Short: 0 <= 1 + 2 and 0 <= 4 + 2 and 0 <= 1 + 3 and 0 <= 4 + 3 (AAxBB).

%t okQ[p_] := Module[{c},

%t c[k_] := c[k] = Count[Mod[p, 5], k];

%t c[0] <= c[1] + c[2] && c[0] <= c[4] + c[2] &&

%t c[0] <= c[1] + c[3] && c[0] <= c[4] + c[3]];

%t a[n_] := a[n] = Count[okQ /@ IntegerPartitions[n], True];

%t Table[Print[n, " ", a[n]]; a[n], {n, 0, 54}] (* _Jean-François Alcover_, Oct 11 2024 *)

%K nonn

%O 0,3

%A _Olivier Gérard_