%I #10 Dec 14 2022 10:56:13
%S 1,1,2,3,6,7,14,15,32,31,63,56,142,101,240,211,467,297,985,490,1524,
%T 1247,2542,1255,6371,1979,7486,7070,14128,4565,32953,6842,42229,37863,
%U 56266,17887,192914,21637,145820,197835,371853,44583,772740,63261,943966,1124840
%N Number of finite sets of integer partitions with all equal sums and total sum n.
%F a(n) = Sum_{d|n} binomial(A000041(d),n/d).
%e The a(1) = 1 through a(6) = 14 sets:
%e {(1)} {(2)} {(3)} {(4)} {(5)} {(6)}
%e {(11)} {(21)} {(22)} {(32)} {(33)}
%e {(111)} {(31)} {(41)} {(42)}
%e {(211)} {(221)} {(51)}
%e {(1111)} {(311)} {(222)}
%e {(2),(11)} {(2111)} {(321)}
%e {(11111)} {(411)}
%e {(2211)}
%e {(3111)}
%e {(21111)}
%e {(111111)}
%e {(3),(21)}
%e {(3),(111)}
%e {(21),(111)}
%t Table[If[n==0,1,Sum[Binomial[PartitionsP[d],n/d],{d,Divisors[n]}]],{n,0,50}]
%o (PARI) a(n) = if (n, sumdiv(n, d, binomial(numbpart(d), n/d)), 1); \\ _Michel Marcus_, Dec 14 2022
%Y This is the constant-sum case of A261049, ordered A358906.
%Y The version for all different sums is A271619, ordered A336342.
%Y Allowing repetition gives A305551, ordered A279787.
%Y The version for compositions instead of partitions is A358904.
%Y A001970 counts multisets of partitions.
%Y A034691 counts multisets of compositions, ordered A133494.
%Y A098407 counts sets of compositions, ordered A358907.
%Y Cf. A000005, A000041, A038041, A055887, A063834, A074854, A289078, A304961, A305552, A306017.
%K nonn
%O 0,3
%A _Gus Wiseman_, Dec 14 2022