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%I #36 Jun 26 2022 14:03:25
%S 1,1,2,2,4,2,5,2,7,3,5,2,13,2,5,4,11,2,13,2,12,4,5,2,28,3,5,5,12,2,18,
%T 2,17,4,5,4,44,2,5,4,24,2,18,2,12,10,5,2,63,3,9,4,12,2,34,4,24,4,5,2,
%U 67,2,5,10,27,4,18,2,12,4,14,2,120,2,5,7,12,4,18,2,54
%N Number of partitions of n in which the sequence of the sum of the same summands is constant.
%C Said differently, these are partitions whose run-sums are all equal. - _Gus Wiseman_, Jun 25 2022
%H Seiichi Manyama, <a href="/A304442/b304442.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) >= 2 for n > 1.
%F a(n) = Sum_{d|n} binomial(A000005(n/d), d) for n > 0.
%e a(72) = binomial(d(72),1) + binomial(d(36),2) + binomial(d(24),3) + binomial(d(18),4) + binomial(d(12),6) = 12 + 36 + 56 + 15 + 1 = 120, where d(n) is the number of divisors of n.
%e --+----------------------+-----------------------------------------
%e n | | Sequence of the sum of the same summands
%e --+----------------------+-----------------------------------------
%e 1 | 1 | 1
%e 2 | 2 | 2
%e | 1+1 | 2
%e 3 | 3 | 3
%e | 1+1+1 | 3
%e 4 | 4 | 4
%e | 2+2 | 4
%e | 2+1+1 | 2, 2
%e | 1+1+1+1 | 4
%e 5 | 5 | 5
%e | 1+1+1+1+1 | 5
%e 6 | 6 | 6
%e | 3+3 | 6
%e | 3+1+1+1 | 3, 3
%e | 2+2+2 | 6
%e | 1+1+1+1+1+1 | 6
%t Table[Length[Select[IntegerPartitions[n],SameQ@@Total/@Split[#]&]],{n,0,15}] (* _Gus Wiseman_, Jun 25 2022 *)
%o (PARI) a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(n/d), d))); \\ _Michel Marcus_, May 13 2018
%Y Cf. A000005 (d(n)), A304405, A304406, A304428, A304430.
%Y All parts are divisors of n, see A018818, compositions A100346.
%Y For run-lengths instead of run-sums we have A047966, compositions A329738.
%Y These partitions are ranked by A353833.
%Y The distinct instead of equal version is A353837, ranked by A353838, compositions A353850.
%Y The version for compositions is A353851, ranked by A353848.
%Y Cf. A098504, A098859, A275870, A353832, A353847, A353864, A353932.
%K nonn
%O 0,3
%A _Seiichi Manyama_, May 12 2018