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Number of partitions satisfying (cn(2,5) = cn(3,5) = 0 and cn(0,5) <= cn(1,5) and cn(0,5) <= cn(4,5)).
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%I #10 Oct 11 2024 06:56:05

%S 1,1,1,1,2,2,3,3,4,5,7,8,10,11,14,19,22,24,29,35,47,52,59,67,83,104,

%T 117,130,150,178,222,247,276,313,371,448,501,554,629,733,879,972,1080,

%U 1214,1408,1661,1837,2027,2279,2613,3063,3369,3718,4151,4741

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

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

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

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

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

%t c[2] == c[3] == 0 && c[0] <= c[1] && c[0] <= c[4]];

%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 0,5

%A _Olivier Gérard_

%E a(0)=1 prepended by _Jean-François Alcover_, Oct 11 2024