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Number of partitions satisfying (cn(0,5) = 0 and 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 15:16:10

%S 1,0,1,1,1,1,3,2,4,5,6,9,11,12,19,22,30,34,44,54,69,85,103,122,155,

%T 184,227,271,325,388,473,557,674,788,939,1113,1319,1554,1830,2137,

%U 2523,2943,3467,4020,4688,5454,6350,7376,8557,9860,11427,13185,15250,17534

%N Number of partitions satisfying (cn(0,5) = 0 and 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: (0 := 0 and 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[0] == 0 && c[1] <= c[2] &&

%t c[1] <= c[3] && c[4] <= c[2] &&

%t 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 0,7

%A _Olivier Gérard_

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