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Number of partitions satisfying (cn(1,5) <= cn(2,5) and cn(1,5) <= cn(3,5) and cn(4,5) <= cn(2,5) and cn(4,5) <= cn(3,5)).
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%I #10 Oct 10 2024 15:14:35

%S 1,0,1,1,1,2,3,3,5,6,9,12,15,18,26,33,45,52,67,86,111,142,170,207,267,

%T 327,414,492,598,740,909,1110,1333,1588,1940,2339,2831,3363,3997,4790,

%U 5736,6836,8108,9533,11326,13409,15863,18662,21862,25683,30212,35425,41476

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

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

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

%t c[1] <= c[2] && c[1] <= c[3] &&

%t c[4] <= c[2] && c[4] <= c[3]];

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

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 45}] (* _Jean-François Alcover_, Oct 10 2024 *)

%K nonn

%O 0,6

%A _Olivier Gérard_

%E a(0)=1 prepended by _Alois P. Heinz_, Oct 10 2024